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OF  ILLINOIS 

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370 

No.  26-34' 


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Digitized  by  the  Internet  Arciiive 

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http://www.archive.org/details/teachersresponsi31monr 


BULLETIN  NO.  31 


BUREAU  OF  EDUCATIONAL  RESEARCH 
COLLEGE  OF  EDUCATION 


THE  TEACHER'S 

RESPONSIBILITY  FOR  DEVISING 

LEARNING  EXERCISES 

IN  ARITHMETIC 


By 


Walter  S.  Monroe 
Director,  Bureau  of  Educational  Research 

Assisted  by 
John  A.  Clark 

Assistant,  Bureau  of  Educational  Research 


JAN  2  5 


price    50   CENTS 


PUBLISHED  BY  THE  UNIVERSITY  OF  ILLINOIS,  URBANA 

1926 


'Hi 


TABLE  OF  CONTENTS 


PAGE 

Preface 5 

Chapter  I.   The  Immediate  Objectives  of  Arithmetic 7 

Chapter  II.    The  Processes  of  Le.^rning  and  Teaching 26 

Chapter  III.   The  Learning  Exercises  of  Arithmetic 33 

Chapter   I\'.    The   Learning   Exercises   Provided  by   Texts   in 

Arithmetic 46 

Chapter  \'.    The   Teacher's   Responsibility   for   Devising   and 

Selecting  Learning  Exercises  in  Arithmetic.    56 

Appendix  A 65 

Appendix  B 90 


PREFACE 

The  research  reported  in  this  monograph  deals  with  a  very  practi- 
cal problem.  Every  teacher  of  arithmetic  continually  faces  the  problem, 
"What  learning  exercises  should  I  ask  my  pupils  to  do?"  It  is  true  that 
few  teachers  explicitly  formulate  this  question  but  all  must  answer  it. 
Incidentally  it  may  be  noted  that  the  answer  given  is  a  very  potent 
factor  in  determining  the  efficacy  of  the  instruction. 

Attempts  to  answer  questions  that  ask  what  should  be  done  may 
be  designated  as  "complete  research"  to  distinguish  such  investigations 
from  fact-finding  inquiries  which  may  be  called  "auxiliary  research." 
The  work  reported  in  this  monograph  represents  an  attempt  to  carry 
out  a  piece  of  "complete  research."  In  this  endeavor  the  results  of  a 
number  of  "auxiliary"  or  "fact-finding"  studies  have  been  utilized  but 
reference  to  them  has  been  subordinated  to  the  consideration  of  the  two 
basic  problems.  Even  the  report  of  the  analysis  of  ten  series  of  arith- 
metic texts,  which  represents  more  than  2500  hours  of  work.  Is  made 
Incidental  to  the  solution  of  these  problems. 

A  critical  reader  will  probably  be  impressed  by  the  Incompleteness 
of  the  data  needed  for  definite  answers  to  the  two  basic  questions.  This 
condition  Is  due  in  part  to  the  complexity  of  these  apparently  simple 
problems  but  the  available  fact-finding  studies  relating  to  them  furnish 
only  fragments  of  the  data  necessary  for  detailed  solutions.  Many  more 
auxiliary  studies  must  be  made  before  we  can  have  what  Is  commonly 
called  a  scientific  answer  to  the  question,  "What  Is  the  responsibility  of 
a  teacher  of  arithmetic  for  devising  and  selecting  learning  exercises.'" 

It  may  even  occur  to  the  critical  reader  that  an  attempt  to  answer 
this  question  Is  not  justified  at  the  present  time  because  the  answer 
must  be  based  upon  fragmentary  data  and  consequently  judgment  must 
be  introduced  at  many  places.  In  reply  to  this  criticism,  one  may  point 
out  that  every  teacher  Is  forced  to  give  some  answer  to  the  question. 
Furthermore,  If  research  workers  become  aware  of  the  Inadequacy  of 
data  for  dealing  with  such  practical  questions,  it  is  possible  that  they 
may  be  stimulated  to  group  their  fact-finding  studies  about  certain 
fundamental  problems.  The  justification  of  auxiliary  studies  is  based 
upon  the  contributions  they  make  to  the  solution  of  problems  that  ask 
"what  should  be." 

Walter  S.  Monroe,  Director 

Bureau  of  Educational  Research 
May  14,  1926. 

[S] 


THE  TEACHER'S  RESPONSIBILITY  FOR  DEVISING 
LEARNING  EXERCISES  IN  ARITHMETIC 

CHAPTER  I 
THE  IMMEDIATE  OBJECTIVES  OF  ARITHMETIC 

The  problem.  The  basic  problems  to  be  considered  In  this  mono- 
graph are  to  determine  (1)  the  nature  and  extent  of  the  learning  exer- 
cises^ provided  by  texts  in  arithmetic  and  (2)  the  responsibility  of  the 
teacher  for  supplementing  a  text  in  this  respect.  In  order  to  assist  the 
reader  in  arriving  at  a  clear  understanding  of  these  problems  and  to 
provide  a  basis  for  their  consideration,  two  subordinate  questions  are 
treated:  (3)  what  are  the  immediate  objectives  of  arithmetic,  and  (4) 
what  learning  exercises  are  needed  for  the  attainment  of  these  objec- 
tives. The  following  pages  of  this  chapter  present  an  exposition  of  the 
objectives  of  arithmetic  as  a  subject  in  the  elementary  school.  Chapters 
II  and  III  are  devoted  to  a  consideration  of  learning  exercises  and 
their  relation  to  objectives.  The  explicit  treatment  of  the  two  basic 
problems  Is  given  in  Chapters  IV  and  V. 

A  general  statement  of  the  objectives  of  arithmetic.  The  purpose 
of  instruction  in  arithmetic  is  to  engender  in  pupils  the  mental  equip- 
ment needed  for  responding  satisfactorily  to  certain  types  of  quantita- 
tive situations  which  they  will  encounter  in  advanced  school  work  and 
in  life  outside  of  school.  This  "mental  equipment"  is  frequently  called 
"ability  in  arithmetic."  Sometimes  the  plural,  "abilities,"  is  used  to  in- 
dicate that  the  equipment  is  not  a  unitary  thing  but  consists  of  a  large 
number  of  elements,  many  of  which  are  independent  in  the  sense  that 
a  pupil  may  acquire  certain  ones  but  not  others. 

This  general  statement,  like  others  which  epitomize  a  group  of 
concepts,  will  probably  not  have  much  meaning  for  the  reader  until 
the  nature  of  the  "mental  equipment"  and  the  situations  for  which  it  is 
to  provide  responses  are  described  in  some  detail. 

Types  of  arithmetical  ability.  Although  psychologically  all  abili- 
ties have  the  common  characteristic  of  a  "bond"  connecting  a  stimulus 
or  situation  and  a  response,  and  no  sharp  lines  of  demarcation  can  be 


^A  learning  exercise  may  be  thought  of  as  a  request  to  do  something.  Examples 
and  problems  are  prominent  as  learning  exercises  in  arithmetic  but  as  we  shall  show 
later  (page  33)  there  are  other  types. 

These  problems  assume  that  exercises  are  to  be  assigned  by  the  teacher.  See 
page  26. 

[7] 


specified,  it  is  possible  to  identify  three  general  types  of  ability;  (1) 
specific  habits,  (2)  knowledge,  and  (3)  general  patterns  of  conduct. 
The  classification  of  a  particular  ability  may  not  always  be  apparent. 
but  the  recognition  of  these  rubrics  will  assist  the  reader  in  arriving  at 
a  clearer  understanding  of  the  immediate  objectives  to  be  attained  by 
the  teaching  of  arithmetic  in  the  elementary  school. 

Nature  of  specific  habits.  If  we  examine  the  way  in  which  pupils 
who  have  studied  arithmetic  respond  to  certain  types  of  situations,  we 
shall  note  certain  distinguishing  characteristics.  For  example,  if  a  fifth- 
grade  pupil  is  directed  to  write  the  numbers  being  dictated  and  "eighty- 
seven"  is  spoken,  he  writes  the  symbols  "87"  and  does  so  without  think- 
ing, that  is.  automatically  and  mechanically.  When  a  number  symbol 
such  as  "4"  is  brought  to  his  attention,  a  meaning-  immediately  comes 
into  his  mind.  In  other  words  the  response,  "meaning  of  the  symbol  4," 
is  connected  with  visual  apprehension  of  the  symbol  so  that,  when  the 
visual  apprehension  occurs,  the  meaning  comes  into  consciousness  and 
does  so  without  the  pupil  making  any  effort  to  recall.  If  a  sixth-grade 
pupil  is  asked  "what  is  the  product  of  7  X  6r",  42  comes  into  his  mind 
as  a  response.  When  he  is  asked  "how  many  feet  in  a  yard:",  he  an- 
swers "three."  When  such  words  as  add,  product,  divide,  multiply, 
equal,  and  the  like  are  brought  to  his  attention,  either  orally  or  in 
printed  form,  a  meaning  immediately  comes  into  his  mind. 

For  situations  of  the  types  illustrated  in  the  preceding  paragraph, 
one  who  has  been  "educated  in  arithmetic"  possesses  a  ready-made  re- 
sponse and  is  able  to  make  it  fluently,  that  is,  quickly  and  with  a  mini- 
mum of  conscious  effort.  Such  mental  equipment  is  usually  designated 
as  motor  skills,  fixed  associations  and  memorized  facts,  or  more  briefl}' 
as  specific  habits.  The  word  "specific"  is  used  to  indicate  that  each  re- 
sponse is  connected  with  only  one  situation  and  that  any  given  situation 
requires  a  certain  response. 

The   scope   of   specific   habits   in   the   field   of   arithmetic.    The 

"tables,"  addition,  subtraction,  and  so  forth  represent  a  number  of  facts 
that  are  to  be  memorized  but  the  total  number  of  specific  habits  in  the 
field  of  arithmetic  is  much  larger  than  is  commonly  realized.  Investi- 
gation has  shown  that  6  -\-  7  and  7  +  6  form  the  basis  of  two  specific 
habits  instead  of  a  single  one.  Additional  specific  habits  are  required 
for  17  +  6  and  16  +  7,  27  +  6  and  26  +  7,  36  +  6  and  36  +  7.  and  so 


^There  are  three  general  types  of  meanings  for  number  symbols:  a  name  for  a 
group  of  objects,  a  position  in  the  number  series  and  its  ratio  to  other  numbers.  The 
meaning  which  a  person  associates  with  a  particular  number  symbol  may  be  a  combi- 
nation of  these  elemental  meanings. 

[8] 


forth.  One  investigator,^  after  careful  inquiry,  has  concluded  that  there 
are  412  addition  combinations  which  a  child  "is  almost  sure  to  need 
after  he  leaves  school."  This  means  that,  if  a  child  fails  to  learn  any 
one  of  these  412  combinations,  there  will  be  a  corresponding  "gap"  in 
his  ability  to  add  integers. 

The  specific  habits  which  function  in  arithmetical  calculations  have 
been  mentioned  first,  but  they  do  not  constitute  all  of  this  type  of  men- 
tal equipment.  Pupils  are  expected  to  learn  the  meaning  of  number 
symbols,  both  integers  and  fractions.  By  implication  this  includes  un- 
derstanding the  decimal  system  of  notation  and  the  ability  to  read  and 
write  numbers  in  Arabic  notation.  In  addition  to  the  symbols  used  in 
expressing  numbers,  the  pupil  is  expected  to  learn  several  such  as  $, 
X,  ^,  and  a  large  number  of  technical  and  semi-technical  terms  includ- 
ing their  abbreviations.  The  topic  of  denominate  numbers  furnishes  a 
large  group  of  such  terms,  but  there  are  many  others  such  as  sum, 
product,  remainder,  percent,  interest,  profit,  loss,  balance,  overdraft, 
discount,  average,  bill,  rectangle,  triangle,  buy,  sell,  at  the  rate  of,  and 
per  yard.  Quantitative  relationships  such  as  the  number  of  feet  In  a 
yard,  number  of  quarts  in  a  gallon,  the  fractional  equivalent  of  12%%, 
and  the  like  furnish  the  basis  for  another  group  of  specific  habits. 

The  nature  of  abilities  designated  as  knowledge.  Specific  habits 
provide  controls  of  conduct  for  responding  to  familiar  situations.  When 
such  situations  are  encountered,  the  pupil  "remembers"  the  responses 
he  found  to  be  satisfactory  on  previous  occasions.  When  "new"  situa- 
tions* are  encountered,  a  pupil's  specific  habits  are  inadequate  as  con- 
trols of  conduct.  He  must  manufacture  a  response  using  the  ideas, 
facts,  concepts,  and  principles  that  the  "new"  situation  suggests  to  him. 
This  mental  equipment  is  called  knowledge^  and  the  process  of  using  It 
is  designated  as  reasoning  or  reflective  thinking. 

It  Is  not  possible  to  specify  a  sharp  line  of  demarcation  between 
specific  habits  and  knowledge.  The  connection  between  a  meaning, 
concept,  or  principle,  and  a  given  situation  may  be  fixed  through  repe- 
tition so  that  the  control  of  conduct  Is  changed  from  knowledge  to  a 
specific  habit.  The  degree  of  the  strength  of  the  bond  connecting  a 
response  with  a  situation  is,  however,  not  the  most  significant  basis  of 


^OsBURN,  W.  J.  Corrective  Arithmetic.  Boston:  Houghton  Mifflin  Company, 
1924,  p.  21. 

*A  "new"  situation  may,  and  frequently  does,  involve  familiar  elements  but  the 
total  combination  is  one  to  which  the  pupil  has  not  responded  before  or  one  for  which 
he  has  forgotten  the  response. 

'^This  definition  of  knowledge  indicates  a  more  restricted  meaning  than  is  com- 
jnonly  associated  with  the  term. 

[9] 


distinction  between  specific  habits  and  knowledge.  The  latter  rubric  of 
controls  of  conduct  is  characterized  by  many  associations  which  result 
in  "richness  of  meaning,"  and  by  organization  which  ties  together  the 
items  of  knowledge  so  that  the  recall  of  one  item  will  tend  to  bring 
related  items  in  one's  mind.  For  example,  assume  that  the  following 
problem  is  "new"  to  a  sixth-grade  pupil:  "The  expenses  of  running  a 
grocery  store  amount  to  20%  of  the  receipts.  How  much  must  a  grocer 
charge  for  a  barrel  of  flour  which  costs  him  $15.00  in  order  to  make  a 
net  profit  equal  to  10%  of  the  selling  price.'" 

Although  this  problem  is  "new"  to  the  pupil,  it  involves  familiar 
elements  (words  and  phrases)  to  which  he  responds  by  recalling  ideas, 
concepts,  and  principles.  These  in  turn  may  suggest  other  items  of 
knowledge.  Out  of  the  total  ideas,  concepts,  and  principles  that  are 
active  in  his  consciousness,  the  pupil  formulates  a  tentative  response 
to  the  problem.  This  is  tested  and  if  found  unsatisfactory  another 
formulation  is  made  and  tested.  The  solving  of  the  problem  is  char- 
acterized by  deliberation  rather  than  fluency.  This  description  of 
knowledge  does  not  constitute  a  detailed  definition  but  it  is  sufficient  for 
our  present  purpose,  which  is  to  point  out  that  knowledge  is  included 
in  the  outcomes  of  arithmetical  instruction. 

General  patterns  of  conduct  as  mental  equipment.  Specific  habits 
and  knowledge  do  not  sufl&ce  as  categories  for  classifying  all  abilities 
resulting  from  the  study  of  arithmetic.  Neatness,  accuracy,  systematic 
attack,  persistence,  and  the  like  designate  controls  of  conduct  which  are 
sometimes  called  habits  or  general  habits.  However,  they  differ  in  sig- 
nificant respects  from  "specific  habits"  described  earlier.  Accuracy 
or  "habit  of  accuracy"  does  not  designate  a  response  to  a  particular 
situation.  It  is  rather  a  general  mode  or  pattern  of  response  to  many 
situations.  Neatness  in  calculating  is  not  a  response  to  a  particular  ex- 
ample such  as  "divide  846.84  by  396"  but  rather  a  general  pattern  of 
response  in  performing  all  calculations.  In  order  to  emphasize  this  dis- 
tinction, the  name  "general  pattern  of  conduct"  is  given  to  such  con- 
trols of  conduct  as  neatness,  accuracy,  and  so  forth. 

Another  aspect  of  the  aim  of  arithmetic.  The  preceding  discussion 
has  pointed  out  three  types  of  mental  equipment  which  teachers  of 
arithmetic  are  expected  to  engender;  first,  specific  habits  that  function 
In  making  calculations  and  In  responding  to  certain  other  types  of  sit- 
uations; second,  knowledge  out  of  which  pupils  will  be  able  to  construct 
responses  to  "new"  problems  which  they  w^Ill  encounter  in  other  school 
activities  and  In  life  outside  of  school;  and  third,  general  patterns  of 
conduct.    This  analysis  of  "ability  in  arithmetic"  has  added  meaning  to 

[10] 


the  general  statement  of  aim  with  which  we  began,  but  the  types  of  sit- 
uations for  which  pupils  are  to  be  equipped  by  the  study  of  arithmetic 
have  been  indicated  only  in  very  general  terms.  A  complete  understand- 
ing of  the  immediate  objectives  of  arithmetic  requires  specifications  in 
regard  to  what  specific  habits,  what  items  of  knowledge  and  what  gen- 
eral patterns  of  conduct  arc  to  be  engendered  by  the  instruction  in  this 
school  subject,  and  the  quality  of  each  ability.*' 

Determination  of  the  particular  arithmetical  abilities  to  be  engen- 
dered. A  method  of  determining  the  specific  habits,  items  of  knowledge 
and  general  patterns  of  conduct  that  should  be  engendered  by  instruc- 
tion in  arithmetic  is  suggested  by  the  statement  of  the  general  purpose 
given  on  page  7.  If  this  instruction  is  to  engender  the  Qmental  equip- 
ment needed  for  responding  satisfactorily  to  certain  types  of  quantita- 
tive situations'?  which  will  be  encountered  in  advanced  school  work  and 
in  adult  life,  it  appears  logical  to  analyze  advanced  school  work  and 
adult  life  for  the  purpose  of  determining  the  quantitative  situations 
involved.'  With  this  information  at  hand,  additional  analyses  should 
reveal  the  nature  and  extent  of  the  arithmetical  equipment  needed  for 
making  satisfactory  responses. 

A  number  of  analyses  of  adult  activities  have  been  made.  One  of 
the  most  elaborate  is  by  Wilson®  who  collected  from  adults  14,583  arith- 
metical problems  which  they  had  encountered  in  their  activities.  From 
his  analysis  of  these  problems  Wilson  reached  certain  conclusions  rela- 
tive to  the  arithmetical  equipment  that  adults  need.  For  example,  the 
demand  for  the  equipment  engendered  by  the  study  of  the  following 
topics  is  so  slight  that  he  recommends  their  elimination: 

1.  Greatest  common  divisor  and  least  common  multiple  beyond 
the  power  of  inspection. 

2.  Long,  confusing  problems  in  common  fractions. 

3.  Complex  and  compound  fractions. 

4.  Reductions  in  denominate  numbers. 

5.  Table    of    folding    paper,    surveyors    table,    tables    of    foreign 
money. 

6.  Compound  numbers,  neither  addition,  subtraction,  multiplica- 
tion nor  division. 

7.  Longitude  and  time. 


*The  quality  of  arithmetical  abilities  is  considered  on  page  IS. 
'This  method  of  determining  educational  objectives  is  called  "job  analysis." 
*WiLsoN,  G.  M.    ''The  social  and  business  usage  of  arithmetic."    Teachers  College 
■Contributions   to   Education,   No.    100.    New  York:    Teachers   College,   Columbia   Uni- 
versity, 1919.    62  p. 

[11] 


8.  Cases  2  and  3  in  percentage. 

9.  Compound  interest. 

10.  Annual  interest. 

11.  Exchange,  neither  domestic  nor  foreign. 

12.  True  discount. 

13.  Partnership  with  time. 

14.  Ratio,  beyond  the  ability  of  fractions  to  satisfy. 

15.  Most    of    mensuration, — the    trapezoid,    trapezium,    polygons, 
frustum,  sphere. 

16.  Cube  root. 

17.  The  metric  system. 

Wilson's  study  also  yielded  information  relative  to  the  character  of 
the  calculations  made  by  adults  in  solving  the  problems  they  encounter. 
Slightly  more  than  half  of  the  additions  involved  either  one  or  two 
place  addends  and  less  than  two  percent  involved  addends  of  more  than 
four  places.  An  analysis  of  a  portion  of  the  problems  showed  that 
nearly  a  third  (31.2  percent)  of  the  additions  involved  only  two  addends 
and  that  less  than  seven  percent  involved  more  than  six  addends.  Sub- 
tractions, multiplications  and  divisions  were  also  shown  to  be  relatively 
simple.  There  were  only  1,974  occurrences  of  common  fractions  in  the 
14,583  problems  and  ten  different  fractions  accounted  for  in  95.5  per- 
cent of  the  cases. ^ 

In  summarizing  his  conclusions  Wilson  states:  "If  to  the  four  fun- 
damentals and  fractions  one  were  to  add  accounts,  simple  denominate 
numbers,  and  percentage,  little  would  be  left  for  all  the  other  processes, 
— so  little  in  fact  that  it  seems  unfair  to  give  attention  to  them  as  drill 
processes  in  the  elementary  schools.  Some  of  them  should  receive  no 
attention.  Others  should  receive  attention  only  for  informational  pur- 
poses or  when  found  necessary  in  the  development  of  motivated  sit- 
uations." 

Limitations  of  the  job-analysis  procedure.  Several  other  investi- 
gators employing  similar  methods,^"  have  contributed  information  in  re- 
gard to  the  arithmetical  equipment  which  adults  use  in  their  activities 
and  it  may  appear  that  such  analyses  when  sufficiently  extended  will 


Tliese  fractions  in  order  of  frequency  are  ^/^,  %,  Yi.  %,  73,  %,  70.  %,  %  and  %. 

^"Adams,  H.  W.  "The  Mathematics  Encountered  in  General  Reading  of  News- 
papers and  Periodicals."  Unpublished  JVIaster's  thesis,  Department  of  Education,  Uni- 
versity of  Chicago  (August,  1924).  Reviewed  by  Bobbitt,  Franklin  K.,  in  Elementary 
School  Journal,  25:133-43,  October,  1924. 

Camerer,  Alice.  "What  should  be  the  minimal  information  about  banking.^" 
Third  Report  of  the  Committee  on  Economy  of  Time  in  Education.    Seventeenth  Year- 

[12] 


yield  a  complete  and  dependable  inventory  of  the  arithmetical  equip- 
ment which  our  schools  should  endeavor  to  engender.  However,  the 
job-analysis  method  of  determining  educational  objectives  has  certain 
limitations  which  should  be  noted.  In  the  first  place  the  functioning  of 
arithmetical  equipment  is  not  confined  to  the  solving  of  problems  or  the 
making  of  calculations.  As  one  comprehends  numbers,  names  of  de- 
nominate quantities,  and  other  items  of  arithmetical  terminology  either 
in  listening  to  a  speaker  or  in  reading,  he  is  using  elements  of  his  arith- 
metical equipment.  Furthermore,  not  infrequently  one  has  occasion  to 
estimate  magnitudes  such  as  the  height  of  a  tree,  the  number  of  tons 
in  a  pile  of  coal,  and  so  forth,  and  to  answer  thought  questions  involv- 
ing quantities  but  not  requiring  calculations.  In  both  estimating  mag- 
nitudes and  answering  quantitative  thought  questions,  one  uses  arith- 
metical equipment  along  with  other  controls  of  conduct. 

A  second  point  is  that  the  present  activities  of  adults  do  not  neces- 
sarily include  all  of  the  uses  of  arithmetical  equipment  that  should  be 
made.  For  example,  authorities  urge  that  farmers  keep  a  detailed 
account  of  their  financial  activities;  that  individuals  keep  personal  ac- 
counts; and  that  heads  of  families  plan  a  budget  at  the  beginning  of 
the  year  and  conform  to  it  as  closely  as  possible.  However,  these  activ- 
ities are  not  engaged  in  by  all  persons  to  whom  they  apply.  In  fact  it 
is  doubtful  if  they  are  engaged  in  at  all  generally. 

A  third  point  to  be  noted  is  that  some  activities  requiring  arith- 
metical equipment  are  engaged  in  by  practically  all  adults  but  other 
activities  are  highly  specialized.  For  example,  everyone  has  occasion 
to  count  money  and  to  check  the  making  of  change  by  clerks  and  store- 
keepers. Most  adults  have  a  bank  account  and  should  keep  the  stub 
of  their  check  book.     On  the  other  hand,  relatively  few  adults  engage 


book  of  the  National  Society  for  the  Study  of  Education,  Part  I.  Bioomington,  Illinois: 
Public  School  Publishing  Company,  1918,  p.   18-26. 

MacLear,  Martha.  "Mathematics  in  current  literature,"  Pedagogical  Seminary, 
30:48-50,  March,  1923. 

Mitchell,  H.  Edwin.  "Some  social  demands  on  the  course  of  study  in  arith- 
metic." Third  Report  of  the  Committee  on  Economy  of  Time  in  Education.  Seven- 
teenth Yearbook  of  the  National  Society  for  the  Study  of  Education,  Part  I.  Bioom- 
ington, Illinois:     Public   School  Publishing  Company,    1918,   p.   7-17. 

Noon,  Philo  G.  "The  child's  use  of  numbers,"  Journal  of  Educational  Psychol- 
ogy, 10:462-67,  November,   1919. 

Smith,  Nila  B.  "An  investigation  of  the  uses  of  arithmetic  in  the  out-of-school 
life  of  first-grade  children,"  Elementary  School  Journal,  24:621-26,  April,   1924. 

Wise,  Carl  T.  "A  survey  of  arithmetical  problems  arising  in  various  occupations," 
Elementary  School  Journal,  20:118-36.  October,  1919. 

Woody,  Clifford.  "Types  of  arithmetic  needed  in  certain  types  of  salesman- 
ship," Elementary  School  Journal,  22:505-20,  March,   1922. 

[13] 


TABLE   I.  OCCUPATIONAL   DISTRIBUTION'  OF   PERSONS   TEN   YEARS 
OF  AGE  AND  OVER,   1920  FEDERAL  CENSUS,  AND  DISTRIBU- 
TION OF  PROBLEMS  IN  FOUR  SERIES  OF  ARITHMETICS 
WITH  RESPECT  TO  THE  ACTIVITY  IN  WHICH 
THEY  AROSE 


Occupational  Division 


Percent  of 
population 


Percent  of 

Problems 

in  Four 

Texts 


Agriculture,  forestry  and  animal  husbandry 

Extraction  of  minerals 

Manufacturing  and  mechanical  industries. 

Transportation , 

Trade , 

Public  service  (not  elsewhere  classified) . . . . 

Professional  service , 

Domestic  and  personal  service 

•Clerical  occupations , 

Activities  ot  the  home , 

Personal  activities 

Activities  of  school  children 

Total 


26.3 

2.6 

30.8 

7.4 

10.2 

1.9 

5.2 
8.2 
7.5 


11. 

0, 
10. 

5. 
43. 

3. 


1.3 

0.1 
0.1 

6.0 

17.0 

1.6 


100.1 


in  certain  vocational  activities  that  provide  many  arithmetical  problems. 
In  the  1920  Federal  Census,  572  occupations  and  occupational  groups 
were  used  in  classifying  the  persons  employed  in  gainful  occupations. 
The  distribution  of  persons  of  ten  years  and  over  among  the  nine  occu- 
pational divisions  is  shown  in  Table  I.  The  largest  percent  (30.8)  is 
for  "manufacturing  and  mechanical  industries"  but  a  large  proportion 
of  those  engaged  in  this  division  of  occupations  are  listed  as  laborers  or 
semi-skilled  employees.  An  analysis^^  of  the  problems  of  four  series  of 
arithmetic  texts  with  respect  to  their  source  gave  the  distribution  shown 
in  the  last  column  of  Table  I.  Obviously,  "trade"  is  the  principal  source 
of  problems  although  it  is  engaged  in  by  only  about  one  person  in  ten. 
In  a  more  detailed  table  that  is  not  reproduced  here,  it  is  shown  that  in 
1910  approximately  55  percent  of  our  population  of  ten  years  of  age 
and  over  were  engaged  in  occupations  to  which  no  arithmetical  prob- 
lems found  in  the  texts  examined  could  be  assigned. 

The  facts  presented  in  Table  I  suggest  that  the  analysis  of  occu- 
pational activities  for  the  purpose  of  identifying  the  arithmetical  prob- 


''MoxROE,  W.\LTER  S.  ''A  preliminary  report  of  an  investigation  of  the  economy 
of  time  in  arithmetic."  Second  Report  of  the  Committee  on  Minimal  Essentials  in 
Elementary  School  Subjects.  Sixteenth  Yearbook  of  the  National  Society  for  the  Study 
of  Education,  Part  I.  Bloomington,  Illinois:  Public  School  Publishing  Company,  1917, 
p.  111-27. 


[14] 


lems  that  occur  has  a  very  limited  value.  The  need  for  the  arithmetical 
equipment  necessary  to  meet  situations  arising  in  particular  occupations 
may  be  greater  than  these  facts  indicate.  When  considering  educational 
•objectives  one  should  recognize  that  the  general  public  may  be  con- 
sidered as  sustaining  a  "consumer's"  relation  to  a  number  of  occupa- 
tions. For  example,  only  one  tenth  of  our  adult  population  is  engaged 
in  trade  occupations  but  practically  everyone  engages  in  buying  and 
therefore  has  occasion  to  check  sale's  slips,  count  change,  and  so  forth. 
It  is  not  possible  for  us  to  know  in  advance  just  which  children  will 
become  clerks,  which  ones  farmers,  which  ones  stenographers,  which 
ones  machine  operators,  and  so  forth.  Furthermore,  persons  engaged 
in  one  occupational  activity  should  know  something  about  the  work  of 
others.  Not  only  is  there  considerable  transfer  of  workers  from  one 
occupation  to  another,  but  social  solidarity  requires  mutual  under- 
standing and  respect,  and  the  more  the  workers  in  one  occupation  know 
of  other  occupations  the  greater  will  be  their  capacity  for  understanding 
and  respecting  their  fellowmen. 

Conclusion  in  regard  to  what  arithmetical  abilities  should  be  en- 
gendered. The  considerations  just  noted  suggest  that  job-analysis 
studies  are  not  likely  to  yield  precise  and  complete  determinations  of 
the  particular  abilities  to  be  engendered  by  instruction  in  arithmetic. 
Studies  already  made  indicate  the  elimination  of  certain  abilities  form- 
erly included  among  the  objectives  of  arithmetic.  Other  studies  have 
indicated  the  inclusion  of  new  abilities  or  increased  emphasis  on  cer- 
tain abilities  already  included.  Future  studies  will  probably  contribute 
to  still  further  refinements  of  arithmetical  objectives  but  the  limitations 
noted  should  not  be  overlooked.  For  the  present  we  are  able  to  compile 
an  inventory  only  in  general  terms  of  the  arithmetical  abilities  to  be 
engendered. 

The  quality  of  arithmetical  equipment.  Another  aspect  of  the  ob- 
jectives of  arithmetic  relates  to  the  quality  of  the  controls  of  conduct 
to  be  engendered.  In  the  case  of  specific  habits  the  quality  of  an  ability 
is  usually  described  in  terms  of  rate  and  accuracy.  For  example,  if  one 
describes  a  pupil's  ability  to  do  addition  examples  of  a  given  type,  he 
specifies  the  rate  at  which  the  addition  is  done  and  the  degree  of  ac- 
curacy of  the  sums.  The  idea  of  both  rate  and  accuracy  is  frequently 
combined  in  the  single  term  "fluency ."^^ 


^For  a  partial  specification  of  fluency  see: 

Herriott,  M.  E.  "How  to  make  a  course  of  study  in  arithmetic."  University  of 
Illinois  Bulletin,  Vol.  23,  No.  6,  Bureau  of  Educational  Research  Circular  No.  37. 
Urbana;    University  of  Illinois,  1925,  p.  10,  11,  29,  and  37. 

[15] 


Another  phase  of  the  quaHty  relates  to  the  "degree  of  permanency." 
If  we  assume  that  it  is  desirable  for  a  student  to  acquire  a  certain  abil- 
ity, how  long  should  he  be  expected  to  retain  this  ability?  A  pupil  may 
learn  a  denominate  number  relation  or  the  meaning  of  a  technical  term 
well  enough  so  that  he  will  remember  it  for  a  week.  Additional  learn- 
ing will  result  in  his  retention  of  the  control  of  conduct  until  the  end  of 
the  school  year  and  if  the  learning  is  continued  sufficiently  the  control 
of  conduct  will  tend  to  become  a  relatively  permanent  acquisition. 

A  description  of  the  specific  habit  and  knowledge  objectives  of 
arithmetics^  The  preceding  discussion  has  indicated  the  difficulties  en- 
countered in  preparing  a  complete  and  detailed  inventory  of  the  spe- 
cific habits  and  items  of  knowledge  to  be  engendered  by  instruction  in 
arithmetic.  As  yet  our  information  concerning  the  demands  for  arith- 
metical equipment  is  so  limited  that  such  an  inventory  cannot  be  form- 
ulated. However,  it  is  possible  to  describe  in  some  detail  the  types  of 
situations  which  children  and  adults  encounter  and  to  indicate  the  na- 
ture of  the  response  to  be  made.  Although  such  a  description  will  be 
subject  to  the  limitation  that  the  range  of  situations  within  each  type 
is  not  determined  except  in  a  general  way,  the  enumeration  of  types 
should  lead  the  reader  to  enrich  his  concept  of  the  objectives  of  arith- 
metic. No  attempt  is  made  to  specify  the  quality  (fluency  or  per- 
mancy)  of  the  abilities  necessary  for  satisfactory  responses,  nor  to  dis- 
tinguish between  specific  habits  and  knowledge. 

I.  Number    symbols.      These    include    Arabic    symbols    0,    1,    2, 

3 9;  numbers  expressed  in  the  decimal  notation,  10,  11,  12 

100,  101,  102,   1000,   1001,   ;  common  fractions  and 

mixed  numbers;  decimals  .5,  .05,   12.5,  .66%,  .6666 ;  Roman 

numerals;  and  verbal  expressions  of  numbers  (both  oral  and  written), 
zero,  one,  two,  one  hundred,  one-half,  first,  second,  and  so  forth. 

Responses  to  be  connected  with  number  symbols.  The  outstand- 
ing type  of  response  to  be  made  to  number  symbols  is  designated  in  a 
general  way  by  the  term  "meaning"  but  the  nature  of  this  response 
need  not  be  always  the  same  for  the  same  type  of  symbol.  For  "small" 
integers  the  meaning  should  sometimes  include  an  image  of  a  definite 
group  of  objects  or  a  rather  precise  idea  of  the  relations  of  the  integer 
to  other  integers.  In  the  case  of  "large"  numbers  a  less  definite  mean- 
ing is  expected.  Usually  a  picture  or  idea  of  the  position  of  a  number 
in  the  number  system  provides  an  adequate  control  of  conduct.  Chil- 
dren should   learn  "definite"  meanings   for  the  more   commonly   used 


'General  patterns  of  conduct  are  considered  on  page  23. 

[16] 


fractions,  both  common  and  decimal,  and  "less  definite"  ones  for  the 
fractions  that  are  encountered  infrequently.  Meaning  responses  should 
be  connected  with  the  Arabic  number  symbols,  verbal  expressions  of 
numbers,  oral,  printed  or  written,  and  with  the  more  commonly  used 
Roman  numerals. 

The  conventional  oral  responses  should  be  connected  with  the  vis- 
ual apprehension  of  the  various  printed  or  written  number  symbols  and 
the  conventional  written  responses  with  the  auditory  apprehension  of 
spoken  number  symbols.  These  two  types  of  equipment  are  required 
for  reading  numbers  and  for  writing  them  from  dictation.  Another 
type  of  equipment  is  needed  for  copying  numbers. 

II.  Other  arithmetical  symbols  and  technical  terms.  In  addition 
to  number  symbols,  certain  conventional  signs  such  as  -f  ?  —  ?  X  ,  -^ , 
==  ,  and  %  are  employed  in  arithmetic.  Closely  related  to  these  are 
the  conventional  arrangements  of  the  number  symbols  indicating  calcu- 
lations. For  example,  numbers  written  in  a  column  with  the  right  hand 
margin  even  indicate  addition.  The  following  arrangement  indicates 
the  division  of  576  by  36. 

16 
36)576 

^e_ 

216 
216 

The  technical  terms  include  (1)  those  relating  to  calculation  such 
as  add,  multiply,  sum,  remainder,  quotient,  partial  product,  "times"  as 
in  6  times  7,  "of"  as  in  %  of  8,  total,  and  average;  (2)  names  of  de- 
nominate numbers  and  their  abbreviations  such  as  quart,  foot,  pound, 
barrel,  dollar,  and  so  forth;  (3)  terms  relating  to  quantitative  aspects 
of  certain  adult  activities  such  as  account,  interest,  balance,  amount, 
change,  profit,  premium,  rectangle,  circle,  and  area;  (4)  words  and 
phrases  such  as  how  many,  how  much,  and,  each,  remains,  bought,  sells, 
lost,  earns,  what  is,  find  (the  sum,  product,  etc.),  and  the  like.  The 
terms  in  the  fourth  group  are  not  peculiar  to  arithmetic  but  in  prob- 
lems they  frequently  have  a  technical  meaning.  Sometimes  they  are 
designated  as  semi-technical  terms. 

Responses  to  be  connected  with  "other  arithmetical  symbols  and 
technical  terms."  The  outstanding  type  of  response  to  be  made  to  this 
class  of  situations  and  stimuli  may  also  be  designated  as  "meaning" 
but  often  the  meaning  of  symbols  and  terms  relating  to  calculation  is 

[17] 


evidenced  by  a  motor  response  as  in  "Find  the  product  of  2894  and 
672."  The  child  is  expected  to  respond  to  this  situation  by  writing 
2894 


,„^  and  not  672)2894  or  some  other  arrangement  of  the  numbers. 
6/2  ^  ^ 


III.  Two  or  more  numbers  quantitatively  related  with  one  miss- 
ing. The  simplest  situations  under  this  head  are  commonly  designated 
as  the  "tables"  or  "basic"  combinations  such  as9-|-3^     ,  7  -\-  0  =     , 

5  (add)   8  (add)  9   (subtract) 

6  0  7-3=    ,     5-0=    ,     4  9X5=    , 

8  (multiply) 

4  12-^4=     ,  9)36.     Until   recently  it  has  been  assumed 

that  the  100  addition  combinations^*  represented  all  of  the  addition 
situations  involving  only  integers  for  which  responses  should  be  mem- 
orized. It  now  appears  that  the  addition  situation  6  4-7  cannot  be  con- 
sidered as  essentially  the  same  as  16  -[-  7,  26  +  7,  and  so  forth,  and 
therefore  300  or  more  "higher  decade"  combinations  must  be  added  to 
the  100  "basic"  combinations.  A  limited  number  of  "higher  decade" 
subtraction  combinations  occur  in  short  division.  There  are  no  addi- 
tional combinations  in  multiplication.  A  feature  of  the  "higher  decade" 
combinations  in  addition  and  subtraction  is  that  one  of  the  numbers 
may  be  an  "inner  stimulus,"  an  idea,  and  not  something  seen  or  heard. 
For  example,  in  adding  the  column  of  figures  shown  on  the  right,  one 
sees  the  7  and  3  but  as  he  adds  up  the  column  he  does  not  see  the 
partial  sums  10,  15,  24,  and  31  to  which  the  numbers  5,  9,  7  8 
and  8  respectively  are  to  be  added.  In  division  it  appears  likely  7 
that  responses  should  be  "learned"  for  all  situations  having  as  9 
divisors  1,  2,  3,  4,  5,  6,  7,  8,  or  9  and  dividends  ranging  from  0  5 
to  those  that  are  10  times  the  divisor. ^^  This  means  that  3 
17-f-2^  ,  13-^4^  ,  and  52^-9^  ,  represent  combina-  7 
tions  as  well  as  16^-2=  ,12-^4=  and  54 -^9=  .  \\'hen 
considered  in  this  way,  division  affords  360  additional  combinations  and 
each  one  can  be  expressed  in  two  ways  such  as  17  -4-  2  =  ,  and  2)  17.^** 
In  addition  to  the  three-number  relationships  described  in  the  pre- 
ceding paragraphs,  there  are  a  number  of  situations  involving  only  two 


^■"Investigation  has  revealed  that  6  +  3  =  and  3  -f-  6  ^  cannot  be  considered 
identical  situations.  Similar  statements  can  be  made  with  reference  to  subtraction, 
multiplication,  and  division.  Hence  there  are  100  "basic"  combinations  in  addition, 
subtraction  and  multiplication,  and  90  in  division. 

"OsBURX,  W.  J.  Corrective  Arithmetic.  Boston:  Houghton  Mifflin  Company, 
1924,  p.  20. 

"The  combinations  described  in  this  paragraph  may  be  called  "secondary"'  to  dis- 
tinguish them  from  the  '"basic"  ones  commonly  referred  to  as  ''the  tables." 

[18] 


numbers;  equivalent  fractions,  such  as  y2  =  %,  %  ^  %>  -5  =  •!/2> 
3^  =  .12%  and  denominate  number  relations  such  as  3  ft.  ==  1  yd., 
1  bu.  =  2150.42  cu.  in. 

Responses  to  be  connected  with  quantitative  relationships  with 
one  number  missing.  The  outstanding  response  to  be  made  to  quanti- 
tative situations  of  the  types  described  in  the  preceding  paragraphs  is 
the  supplying  ot  the  missing  number.  Sometimes  this  response  is  to 
be  expressed  in  written  form;  on  other  occasions  it  is  to  be  partially 
written  (e.  g.,  units  written  and  tens  carried);  and  in  column  addition 
and  multiplication  the  response  may  not  be  expressed  but  functions  as 
an  element  in  the  next  situation. 

IV.  Examples.  The  term  "examples"  is  used  to  designate  explicit 
requests  to  add,  subtract,  multiply,  divide  or  extract  a  root,  the  num- 
bers being  given.  Examples  differ  from  "requests  for  the  missing  num- 
ber in  a  specific  quantitative  relationship"  in  respect  to  the  manner  in 
which  the  response  is  given.  In  the  latter  class  of  exercise,  the  pupil  is 
expected  to  memorize  the  response  and  when  two  numbers  are  given  he 
is  expected  to  "remember"  the  response.  There  is  no  calculation.  In 
the  case  of  examples,  the  pupil  responds  to  elements  of  the  request  and 
builds  up  the  response  to  the  total  situation.  In  addition  to  those 
given  in  explicit  form,  examples  are  created  in  solving  verbal  problems 
(see  page  21)  when  a  decision  has  been  reached  in  regard  to  the  calcu- 
lations to  be  performed. 

Response  to  examples.  A  fluent  response  is  to  be  made  to  ex- 
amples, that  is,  one  needs  to  be  equipped  so  that  he  can  perform  the 
specified  calculation  accurately  and  with  reasonable  speed.  In  making 
this  response  one  utilizes  his  ability  to  respond  to  the  basic  and  sec- 
ondary number  combinations. 

Sometimes  one  is  expected  to  be  able  to  make  a  special  response, 
that  is,  employ  a  short  cut  or  use  a  calculating  device  such  as  an  interest 
table. 

V.  Questions,  usually  implied,  concerning  functional  relation- 
ships. A  question  concerning  a  functional  relationship^^  is  implied  in 
the  statement  of  a  problem.  Consider  the  problem,  "If  a  quart  of 
paint  covers  9  sq.  yd.  of  floor  surface,  how  much  paint  is  required  for 
the  floor  of  a  porch  12  ft.  by  20  ft.:"  This  problem  implies  the  question, 
"How  is  the  area  of  the  porch  floor  in  square  yards  to  be  calculated 
from  the  dimensions  12  ft.  and  20  ft?"    The  problem,  "Find  the  value 


"A  functional  relationship  is  a  statement  of  the  quantitative  relation  between 
certain  general  quantities  such  as  base,  altitude  and  area  of  a  rectangle,  or  the  face  of 
a  note,  time,  rate  of  interest,  and  amount  due. 

[19] 


of  52.3  bu.  of  wheat  at  $1.27  per  bushel,"  imphes  the  question,  "How 
is  the  value  of  a  number  of  units  (in  this  case  52.3  bu.)  calculated 
when  the  number  of  units  and  price  per  unit  are  given?" 

A  search  through  the  literature  relating  to  the  objectives  of  arith- 
metic has  failed  to  reveal  any  attempt  to  determine  the  particular  ques- 
tions concerning  functional  relationship  which  children  should  learn  to 
identify  in  problems  and  problematic  situations  and  to  answer.  In  an  an- 
alysis of  the  second  and  third  books  of  ten  three-book  series  of  arithme- 
tics 333  types  of  questions  were  recognized.^*  A  few  illustrations  are 
given  here  and  a  suggested  minimum  essential  list  is  given  as  Appendix 
B.  page  90. 

What  calculation  must  be  made: 

To  find  the  total,  given  two  or  more  items,  values,  and  so  forth. 

To  find  the  amount,  or  number  needed,  given  a  magnitude  and  the  number  of 
times  it  is  to  be  taken. 

To  find  how  many  when  reduction  ascending  is  required,  given  a  magnitude 
expressed  in  terms  of  two  or  more  denominations. 

To  find  the  total  price,  given  the  number  of  units  and  the  price  per  unit  of 
another  denomination. 

To  find  the  return  percent  on  an  investment,  given  the  net  profit  or  net  income 
and  amount  invested. 

To  find  the  rate  of  profit,  given  the  cost  of  goods,  and  the  expenses  and  losses. 

The  reader  should  bear  in  mind  that  verbal  problems  implying  the 
same  question  concerning  a  functional  relation  may  vary  greatly  in  form 
of  statement.  For  example,  the  question,  "How  may  the  number  of 
units  bought  or  sold  be  calculated  when  the  price  or  value  per  unit  and 
the  total  price  or  value  are  known,"  is  implied  in  each  of  the  following 
problems:^" 

1.  How  many  yards  of  silk  at  $1.50  per  yard  can  be  bought  for  $7-50? 

2.  The  silk  for  a  dress  cost  $7.50.  How  many  yards  were  purchased  at  $1.50 
per  yard: 

3.  At  $1.50  per  yard,  how  many  yards  of  silk  does  a  woman  get  if  the  amount 
of  the  purchase  is  $7.50? 

4.  At  the  rate  of  $1.50  per  yard  my  bill  for  silk  was  $7.50.  How  many  yards 
were  purchased? 

5.  How  many  yards  of  silk  at  $1.50  a  yard  does  a  bill  of  $7.50  represent? 

6.  When  silk  is  $1.50  a  yard,  a  piece  of  silk  costs  $7.50.  How  many  yards  in 
the  piece? 

7.  At  $1.50  a  yard  how  many  yards  of  silk  does  a  merchant  sell  if  he  receives 
$7.50  for  the  piece? 

8.  Mrs.  Jones  purchased  silk  at  $1.50  a  yard.  The  entire  amount  paid  was  $7.50. 
How  many  yards  were  bought? 

9.  Silk  was  sold  at  $1.50  per  yard.  A  check  for  $7.50  was  given  in  settlement. 
Find  the  number  of  yards  bought. 

"This  analysis  is  described  on  page  41  and  the  333  types  of  questions  are  given 
in  Appendix  A. 

'*The  problems  of  this  list  were  suggested  by  analysis  of  several  texts. 

[20] 


10.  At  $1.50  per  yard,  how  many  yards  can  be  bought  for  $7.50? 

11.  A  merchant  sells  a  number  of  yards  of  silk  for  $7.50.  The  price  being  $1.50 
for  each  yard,  how  many  does  he  sell? 

12.  I  invested  $7.50  in  silk  at  $1.50  per  yard.     How  many  yards  did  I  buy? 

13.  When  silk  is  $1.50  per  yard,  how  many  yards  can  be  bought  for  $7.50? 

14.  When  silk  is  sold  for  $1.50  for  each  yard,  what  quantity  can  be  bought 
for  $7.50? 

15.  At  the  rate  of  $1.50  per  yard,  how  many  yards  can  be  bought  for  $7.50? 

16.  Silk  is  selling  for  $1.50  per  yard,  how  many  yards  should  be  sold  for  $7.50? 

17.  At  a  cost  of  $1.50  a  yard,  how  many  yards  can  be  bought  for  $7.50? 

18.  Silk  was  bought  at  a  cost  of  $1.50  per  yard.  At  that  rate,  how  many  yards 
can  be  bought  for  $7.50? 

19.  At  $1.50  a  yard  a  piece  of  silk  cost  $7.50.    How  many  yards  in  the  piece? 

20.  How  many  yards  of  silk  at  $1.50  can  I  buy  for  $7.50? 

21.  $7.50  was  paid  for  silk  at  $1.50  per  yard.    How  many  yards  were  bought? 

22.  Find  the  number  of  yards;  cost  $7.50.    Price  per  yard  $1.50. 

23.  The  cost  of  a  piece  of  cloth  is  $7.50  and  the  cost  per  yard  $1.50.  How  many 
yards  are  there  in  the  piece? 

24.  A  woman  paid  $7.50  for  a  piece  of  silk  that  cost  her  $1.50  per  yard.  How 
many  yards  were  there  in  the  piece? 

25.  A  woman  had  $7.50  and  bought  silk  at  $1.50  a  yard.  How  many  yards  did 
she  buy? 

26.  A  quantity  of  silk  at  $1.50  per  yard  cost  $7.50.    What  was  the  quantity? 

27.  Silk  is  $1.50  a  yard  and  I  bought  $7.50  worth  today.  How  many  yards  did 
I  buy? 

28.  A  woman's  bill  for  silk  was  $7.50.  If  each  yard  cost  $1.50,  how  many  yards 
were  bought? 

This  list  does  not  exhaust  the  types  of  statements  of  one-step 
problems  which  imply  this  question.  It  is  also  implied  in  combination 
with  other  questions  in  many  problems  involving  two  or  more  steps. 
However,  the  list  illustrates  something  of  the  variety  of  situations  (prob- 
lem statements)  to  which  the  response  "divide  the  total  price  (cost, 
value,  etc.)  by  the  price  (cost,  value,  etc.)  per  unit  and  the  quotient 
will  be  the  number  of  units"  is  to  be  given. 

Response  to  be  given  to  questions  concerning  functional  relation- 
ships. The  type  of  response  to  be  given  to  questions  concerning  general 
quantitative  relationships  is  described  in  the  preceding  paragraph.  The 
reader  should  note  that  the  answer  specifies  certain  calculations  to  be 
made. 

VI.  Verbal  problems.  Verbal  problems  have  been  referred  to  in 
the  preceding  pages  but  when  they  constitute  occasions  for  manu- 
facturing a  response  by  reflective  thinking  rather  than  recalling  a  ready- 
made  response,  that  is,  when  they  are  "new"  and  are  really  problems 
for  the  pupil,  they  justify  recognition  as  an  additional  type  of  situation 
for  which  arithmetical  equipment  is  needed.  It  is  not  possible  to  give 
an  objective  definition  of  the  line  of  demarcation  between  the  problems 
to  which  one  responds  by  reflective  thinking  and  the  "problems"  that 

[21] 


he  "solves"  by  recalling  a  ready-made  response.  Almost  any  problem 
may  come  under  the  second  type  provided  a  person  encounters  it  or 
very  similar  exercises  sufficiently  frequently  so  that  the  bond  connect- 
ing the  required  response  with  the  situation  represented  by  the  verbal 
statement  of  the  problem  has  become  fixed.  When  this  happens  the 
problem  ceases  to  be  a  "problem"  for  the  person  in  question,  that  is, 
it  is  not  a  situation  requiring  reflective  thinking.  For  example,  a  sev- 
enth-grade pupil  may  think  reflectively  in  solving  an  interest  problem 
but  a  banker  would  respond  to  it  in  much  the  same  way  as  the  pupil 
responds  to  a  request  to  multiply  846  by  52. 

A  verbal  problem  in  the  sense  the  term  is  used  here  is  a  new  situ- 
ation, that  is,  one  for  which  the  person  does  not  have  a  ready-made  re- 
sponse. Thus  when  we  state  that  one  of  the  objectives  of  instruction  in 
arithmetic  is  to  equip  the  pupils  to  solve  verbal  problems,  we  mean  that 
they  are  to  be  equipped  to  respond  satisfactorily  to  situations  to  which 
they  have  not  responded  previously,  that  is,  to  answer  questions  they 
have  not  answered  in  their  study  of  arithmetic. 

A  7iew  situation  is  not  necessarily  new  in  all  its  elements.  In  fact 
the  opposite  is  usually  true.  A  new  problem  will  usually  involve  many 
familiar  words  and  phrases.  The  implied  questions  relative  to  general 
quantitative  relations  will  usually  be  familiar.  The  total  situation,  how- 
ever, is  new  either  because  unfamiliar  elements  are  introduced  or  be- 
cause familiar  elements  appear  In  a  new  combination. 

Response  to  problematic  situations.  As  indicated  in  the  preceding 
paragraphs,  the  nature  of  the  response  one  makes  is  the  distinguishing 
characteristic  of  a  problematic  situation.  The  response  is  complex.  In 
so  far  as  the  situation  is  familiar,  the  elements  of  the  response  belong 
under  other  types  of  situations.  Meanings  are  connected  with  words 
and  symbols;  the  implied  question  concerning  a  functional  relationship 
is  identified  and  answered;  denominate  number  facts  are  recalled;  num- 
bers are  read  and  copied.  However,  the  response  cannot  be  adequately 
described  by  enumerating  the  responses  to  such  elements.  Reflective 
thinking  is  involved.  It  should  be  noted  that  the  total  response  to 
a  verbal  problem  includes  the  determination  of  the  calculations  to  be 
performed  plus  the  response  to  the  example  formulated.  Reflective 
thinking  Is  Involved  In  only  the  first  phase  of  the  total  response. 

VII.  Informational  questions  about  business  and  social  activities. 
Adults  find  occasion  to  answer  a  number  of  Informational  questions  re- 
lating to  such  activities  as  banking,  transportation,  transmitting  money, 
taxation,  Insurance,  manufacturing,  construction,  and  so  forth.  The 
following  are  typical:  How  Is  money  transmitted:    What  is  a  promisory 

[22] 


note?  What  is  a  sight  draft?  How  does  a  city  secure  funds  for  pav- 
ing streets?  How  are  taxes  levied?  What  is  board  measure?  What  is 
overhead?  How  is  postage  computed  on  parcels?  What  conditions 
affect  fire  insurance  rates?  The  questions  may  be  asked  in  explicit 
terms  but  frequently  they  are  implied  in  a  general  request  or  need. 

The  range  of  such  questions  for  which  instruction  in  arithmetic  is 
expected  to  engender  equipment  has  not  been  determined  but  it  is  ob- 
vious that  other  school  subjects,  especially  geography  and  civics,  must 
assume  some  of  the  responsibility  for  equipping  pupils  to  answer  in- 
formational questions  relating  to  business  and  social  activities. 

Response  given  to  informational  questions  relating  to  business 
and  social  activities.  The  general  nature  of  the  response  to  informa- 
tional questions  relating  to  business  and  social  activities  is  implied  by 
the  illustrative  questions  in  the  preceding  paragraph.  However,  it  may 
be  noted  that  usually  precise  and  definite  answers  are  required, 

VIII.  "Practical  experiences."  Under  the  head  of  "practical  ex- 
periences" we  group  a  number  of  types  of  situations  such  as  (1)  United 
States  currency  and  other  collections  of  objects  to  be  counted,  (2) 
magnitudes  to  be  estimated  or  measured  in  terms  of  some  unit,  (3) 
business  forms  (sales  slips,  checks,  money  orders,  etc.),  catalogue  lists, 
proposals  for  bond  issues,  newspaper  quotations,  and  so  forth  to  be 
comprehended,  (4)  situations  in  which  arithmetical  problems  are  to  be 
identified  and  formulated. 

The  engendering  of  the  arithmetical  equipment  required  for  re- 
sponding to  the  situations  enumerated  under  the  head  of  "practical  ex- 
periences" represents  important  objectives.  The  need  for  counting  ob- 
jects, estimating  or  measuring  magnitudes,  and  comprehending  busi- 
ness forms  is  generally  recognized  but  the  need  for  identifying  and  form- 
ulating the  arithmetical  problems  arising  in  practical  situations  is  even 
more  important.  With  few  exceptions  adults  seldom  need  to  solve  a 
verbal  problem  stated  by  another  person.  Their  problems  are  encount- 
ered in  their  "practical  experiences"  and  before  the  solution  is  begun 
the  problem  nmst  be  formulated,  at  least  mentally. 

General  patterns  of  conduct  as  objectives  in  arithmetical  instruc- 
tion. As  stated  on  page  10  a  general  pattern  of  conduct  does  not  provide 
a  response  to  a  particular  situation  but  it  exercises  a  general  control  of 
one's  responses  to  many  situations,  the  range  depending  upon  the  ex- 
tent of  the  generalization  of  the  pattern. 

Accuracy  or  the  "habit  of  accuracy"  is  usually  listed  as  an  objective 
of  arithmetical  instruction  but  it  is  different  from  the  specific  habits 

[23] 


which  function  in  performing  calculations.  The  latter  designate  definite 
responses  to  definite  situations.  A  "habit  of  accuracy"  is  a  general 
pattern  of  conduct  which  controls  responses  to  a  variety  of  situations 
which  in  this  case  are  calculations.  This  control  may  result  in  per- 
forming the  calculation  a  second  time,  checking,  inspecting  the  work 
for  errors,  judging  the  answer  with  respect  to  reasonableness,  and  the 
like.  A  person  who  has  attained  the  "habit  of  accuracy"  tends  to  give 
one  or  more  of  these  responses  to  any  calculation  situation.  The  pres- 
ence of  the  word  "habit"  indicates  that  the  response  always  tends  to 
be  made  and  is  made  skillfully.  Much  the  same  idea  is  expressed  by 
the  statement  that  a  person  who  possesses  the  "habit  of  accuracy" 
knows  what  to  do  in  order  to  attain  accuracy  and  how  to  do  it.  and 
derives  satisfaction  from  doing  what  is  necessary. 

Other  general  patterns  of  conduct  listed  among  the  objectives  of 
arithmetic  are  neatness,  honesty,  initiative  and  resourcefulness  in  solv- 
ing problems,  perseverence,  and  systematic  procedure.  A  general  pat- 
tern of  conduct  which  may  be  designated  as  a  "problem  solving  atti- 
tude" is  implied  in  some  of  the  statements  of  the  aim  of  arithmetic.  Its 
central  element  appears  to  be  the  belief  that  the  way  to  respond  to 
a  new  situation,  that  is,  a  problem,  is  to  ascertain  what  is  known  about 
it  and  precisely  what  question  or  questions  are  to  be  answered,  and  then 
to  focus  one's  resources  upon  the  problem  in  an  attempt  to  manufac- 
ture a  response  by  formulating  solutions  (hypotheses)  and  testing  them 
until  a  satisfactory  one  is  found.  Usually  there  is  coupled  with  this 
belief,  confidence  in  one's  own  ability  to  solve  the  problem.  The  absence 
of  these  phases  of  a  "problem  solving  attitude"  is  evidenced  when  a 
pupil  searches  in  his  text  for  the  solution  of  a  similar  problem  or  re- 
stricts his  efforts  to  recalling  the  solution  of  a  similar  problem. 

Another  significant  phase  of  the  "problem  solving  attitude"  is  in- 
volved in  the  pupil's  concept  of  what  it  means  to  solve  a  problem.  One 
point  of  view  is  that  to  solve  a  problem  is  to  get  the  answer  given  in 
the  text  or  one  that  will  be  accepted  by  the  teacher.  The  "problem 
solving  attitude"  requires  that  one  think  of  the  solving  of  a  problem  as 
a  case  of  reflective  thinking  in  which  the  fundamental  objective  is  to 
conform  to  the  requirements  of  good  thinking. 

Summary.  Although  the  preceding  discussion  of  the  objectives  of 
arithmetic  has  filled  several  pages,  it  has  doubtless  been  apparent  to  the 
reader  that  the  items  of  mental  equipment  (specific  habits,  knowledge 
and  general  patterns  of  conduct)  to  be  engendered  by  the  instruction  in 

[24] 


arithmetic  have  not  been  specified  at  all  completely ."°  However,  the 
enumeration  of  eight  types  of  situations  to  which  pupils  are  to  be 
equipped  to  respond  and  the  consideration  of  general  patterns  of  con- 
duct should  lead  the  reader  to  attach  more  meaning  to  the  first  state- 
ment of  the  purpose  of  instruction  in  arithmetic  (see  page  7).  The 
teaching  of  arithmetic  is  expected  to  engender  the  specific  habits, 
knowledge,  and  general  patterns  of  conduct  needed  for  responding  in  a 
satisfactory  way  to  the  following  general  classes  of  situations: 
I.  Number  symbols. 
II.  Other  arithmetical  symbols  and  technical  terms. 

III.  Two  or  more  numbers  quantitatively  related  with  one  miss- 
ing. 

IV.  Examples. 

V.  Questions    (usually  implied)   concerning  functional  relation- 
ships. 
VI.  Verbal  problems. 
VII.  Informational  questions  about  business  and  social  activities. 
VIII.  Practical  experiences:  (1)  collections  of  objects  to  be  counted, 

(2)  magnitudes    to    be    estimated    In    terms    of    some    unit, 

(3)  business  forms,  catalogue  lists,  newspaper  quotations, 
and  so  forth  to  be  comprehended,  (4)  situations  in  which 
arithmetical  problems  are  to  be  Identified  and  formulated. 


■*The  situations  for  which  abihty  to  respond  should  be  engendered  have  been 
described  only  in  terms  of  types  and  no  attempt  has  been  made  to  specify  the  quality 
of  the  several  abilities. 


[25] 


1/ 


CHAPTER  II 
THE  PROCESSES  OF  LEARNING  AND  TEACHING 

The  discussion  of  the  objectives  of  arithmetic  in  Chapter  I  furn- 
ishes a  description  of  what  pupils  should  learn  during  their  study  of 
this  subject  in  the  elementary  school.  The  problem  of  this  chapter  is 
to  describe  certain  phases  of  the  learning  process  and  the  teaching  pro- 
cedures that  are  essential  to  the  attainment  of  these  objectives.  ^ 

Learning  an  active  process.  In  discussing  the  work  of  the  teacher, 
we  commonly  use  verbs  such  as  "impart,"  "communicate,"  "present," 
"explain,"  and  "instruct,"  which  not  infrequently  appear  to  imply  that 
in  the  process  of  educating  children  the  teacher  transmits  specific  habits, 
items  of  knowledge  or  general  patterns  of  conduct  to  the  pupil  whose 
mind  may  be  receptive  or  even  eager  to  receive  or  may  be  indifferent  or 
hostile.  Although  no  one  who  is  informed  in  regard  to  modern  psy- 
chology would  support  such  a  theory  of  learning,  the  reading  of  current 
educational  literature  and  the  observation  of  classroom  procedures  sug- 
gest that  many  teachers,  in  planning  lessons  and  in  conducting  recita- 
tions, do  assume  that  their  function  is  to  transmit  ideas,  facts,  rules, 
and  other  items  of  knowledge  to  their  pupils. 

The  statement,  "learning  is  an  active  process,"  is  commonplace 
but  its  significance  is  far-reaching.  What  a  child  learns  is  the  product 
of  his  own  activity,  physical,  mental,  and  emotional.  A  child  who  is 
not  active  does  not  learn.  In  order  to  learn  the  multiplication  combi- 
nations a  child  must  enage  in  certain  types  of  activity;^  in  no  other  way 
can  he  acquire  the  necessary  specific  habits  (fixed  associations). 

Assignment  of  exercises  required  as  a  basis  for  attaining  arith- 
metical objectives.  Acceptance  of  the  thesis,  that  learning  is  an  active 
process  and  that  the  acquiring  of  certain  abilities  requires  participation 
in  certain  types  of  learning  activities,  raises  the  question:  What  means 
must  the  teacher  employ  to  secure  the  pupil  activity  that  will  lead  to 
the  attainment  of  the  objectives  described  in  Chapter  I?  A  child  learns 
as  the  result  of  his  activity  outside  of  the  school  such  as  playing  games; 
doing  errands  for  his  parents,  including  the  making  of  purchases;  read- 
ing newspapers,  magazines   and  books;   constructing  toys,   playhouses 


^The  reader  should  not  interpret  this  statement  to  mean  that  all  pupils  must  go 
through  the  same  activities.  The  activities  of  one  pupil  engaged  in  learning  the  multi- 
plication combinations  may  differ  in  certain  respects  from  those  of  another  pupil  work- 
ing toward  the  same  end  but  the  activities  of  the  two  pupils  will  have  certain  common 
characteristics.    For  example,  in  this  case  both  will  involve  repetition. 

[26] 


and  the  like;  being  a  member  of  an  organization  such  as  the  Boy  Scouts 
or  Campfire  Girls;  observing  the  activities  of  adults,  and  the  like.  How- 
ever, the  establishment  of  schools  is  evidence  of  the  recognition  of  the 
fact  that  the  abilities  resulting  from  participation  in  such  activities  dur- 
ing the  period  of  childhood  would  seldom  if  ever  constitute  adequate 
equipment  for  meeting  the  demands  of  adult  life.  If  our  schools  were 
abolished,  many  necessary  abilities  would  not  be  acquired  and  the  qual- 
ity of  others  would  be  unsatisfactory.  Prior  to  the  inclusion  of  arith- 
metic in  the  curriculum  of  the  "public  schools,"  very  few  children  ac- 
quired ability  to  "cipher"  and  those  who  learned  arithmetic  did  so  as 
the  result  of  attending  a  special  school  in  which  it  was  taught. 

Participation  in  efficient  educative  activities  in  school  is  not  secured 
by  the  teacher  making  the  direct  request,  "Be  active"  or  "Do  some- 
thing." In  response  to  such  commands  or  requests  the  pupils  may  be- 
come active.  In  fact  children  are  by  nature  inclined  to  be  active  and 
unless  restrained  will  "do  something,"  but  "spontaneous"  activity  is 
very  unlikely  to  contribute  greatly  to  their  mental  equipment,  at  least 
in  the  field  of  arithmetic.  Observation  of  the  teaching  of  arithmetic 
justifies  the  statement  that  most  of  the  educative  activity  in  this  field  is 
in  response  to  definite  exercises"  assigned  by  the  teacher. 

Those  who  believe  in  the  project  method  will  probably  take  issue 
with  the  thesis  of  the  preceding  paragraph.  It  is  true  that  needs  which 
furnish  a  basis  for  efficient  educative  activities  do  arise  in  the  attempts 
of  pupils  to  realize  their  own  immediate  purposes  both  in  and  out  of 
school.  It  is  also  true  that  the  number  of  such  needs  can  be  increased 
by  skillful  encouragement  and  manipulation  of  conditions  by  the  teacher, 
but  it  does  not  appear  that  in  general  the  project  method  can  be  de- 
pended upon  to  stimulate  the  learning  activities  necessary  to  produce 
the  mental  equipment  specified  by  the  objectives  of  arithmetic.  In  most 
teaching  situations  there  must  be  explicit  assignment  of  exercises  which 
will  function  as  the  basis  for  much,  and  in  many  instances  practically 
all,  of  the  educative  activity  in  the  field  of  arithmetic.^ 

Types  of  learning  activities  resulting  in  the  acquisition  of  specific 
habits  and  knowledge.  Before  consideration  is  given  to  the  types  of 
exercises  assigned  by  teachers  of  arithmetic,  it  will  be  helpful  to  note 
the  types  of  learning  activities  in  which  children  engage  in  acquiring 

"The  types  of  exercises  employed  in  arithmetic  will  be  considered  later  (seepage  33). 

^For  a  more  complete  discussion  of  the  "project  method"  versus  the  "assignment 
method"  see: 

Monroe,  Walter  S.  "Projects  and  the  project  method."  University  of  Illinois 
Bulletin,  Vol.  23,  No.  30,  Bureau  of  Educational  Research  Circular  No.  43.  Urbana: 
University  of  Illinois,  1926.   20  p. 

[27] 


specific  habits  and  knowledge.  In  attempting  to  analyze  mental  activ- 
ity, psychologists  have  identified  such  processes  as  s^nsgljon,  jiexcep- 
tion,  conception,  imagination,  niernory,  association,  analysis,  generaliza- 
tion,  and  reasoning.  Recognition  of  these  mental  processes  is  helpful 
for  certain  purposes  but  a  somewhat  different  analysis  of  learning  ap- 
pears to  provide  a  more  practical  basis  for  considering  the  technique  of 
teaching.  The  "types  of  learning  activity"  enumerated  in  the  follow- 
ing paragraphs  represent  a  pedagogical  rather  than  a  scientific  analysis 
of  the  learning  process. 

1.  Direct  or  perceptual  experiencing  occurs  in  learning  arithmetic 
when  a  pupil  counts  objects,  measures  the  length  of  the  room,  handles 
weights  or  money,  steps  off  a  distance,  and  the  like.  In  direct  experi- 
encing there  is  perception  and  hence  the  functioning  of  one  or  more  of 
the  sense  organs.  Perceptual  experiences  are  required  as  a  basis  for  the 
other  rubrics  of  learning  activity.  Much  of  the  necessary  experiencing 
will  take  place  outside  of  the  school  but  when  additional  experiencing  is 
required  the  teacher  must  provide  opportunity  for  this  type  of  learning 
activity.  Perceptual  experiencing  occurs  in  measuring  and  counting  ob- 
jects, dramatizing  adult  activities,  visiting  business  concerns  such  as 
banks,  grocery  stores,  department  stores,  and  the  like. 

2.  Vicarious  experiencing  occurs  when  one  listens  to  or  reads  an 
account  of  the  perceptual  experiences  of  another,  provided  the  listener 
or  reader  comprehends  the  terms  used  in  the  description.  For  example, 
a  pupil  may  experience  vicariously  or  second-hand,  without  visiting  and 
observing  Its  activities,  the  operation  of  a  building  and  loan  associa- 
tion. When  a  person  observes  an  activity  such  as  an  athletic  contest 
his  experience  Is  a  combination  of  direct  and  vicarious.  His  playing  of 
the  game  is  vicarious  but  his  seeing  of  the  players  and  the  spectators  is 
direct.  A  foundation  of  direct  or  first-hand  experience  Is  an  essential 
prerequisite  for  vicarious  experiencing. 

3.  Generalizing  experience*  is  used  as  a  name  for  analyzing,  com- 
paring, organizing,  and  abstracting  experiences,  both  direct  and  vicari- 
ous. The  products  of  these  activities  are  called  concepts,  rules,  prin- 
ciples, generalizations,  and  abstractions.  Words  like  sum,  multiplica- 
tion, interest,  premium,  volume,  and  fraction  represent  concepts.  They 
are  sometimes  called  abstract  and  general  meanings. 

4.  Comprehending  the  products  of  thought  expressed  in  terms  of 
words,  phrases  or  sentences  designates  a  type  of  learning  activity  which 
In   some   respects    Is   the    reverse   of   generalizing   experience.      In   the 


^''Inductive  development"  has  also  been  used  as  a  name  for  this  type  of  activity. 

[28] 


course  of  the  history  of  the  race  a  number  of  terms  have  been  developed 
for  use  in  describing  arithmetical  calculations  and  in  stating  problems. 
A  pupil  encounters  such  words  as  addition,  multiplication,  interest,  pre- 
mium, numerator,  percent,  and  the  like.  Before  he  has  learned  from 
his  experiencing  the  meaning  they  represent  he  faces  the  necessity  of 
comprehending  or  understanding  the  products  of  the  thinking  of  other 
persons.  A  similar  statement  can  be  made  with  reference  to  rules  and 
principles. 

5.  Using  one's  knowledge  in  manufacturing  a  response  to  a  new 
situation^  is  commonly  called  "problem  solving"  or  reflective  thinking. 
These  terms,  however,  are  used  somewhat  carelessly  and  for  this  reason 
the  writer  has  chosen  a  descriptive  phrase  which  explicitly  specifies  that 
the  learner  is  engaged  in  responding  to  a  new  situation.  It  is  generally 
assumed  that  reflective  thinking  occurs  when  a  pupil  responds  to  a  ver- 
bal problem.  This  is  not  true  because  the  pupil  may  remember  how  the 
problem  or  a  similar  one  was  solved  In  the  text  or  by  the  teacher,  or  he 
may  search  his  text  for  the  solution  of  a  similar  problem.  When  this 
constitutes  his  activity  he  Is  not  thinking  reflectively,  that  Is,  he  Is  not 
manufacturing  a  response  to  a  new  situation.  He  Is  simply  searching 
for  a  ready-made  response.  Random  guessing  represents  another  type 
of  activity  that  Is  not  Included  under  the  caption  used  here,  "Using 
knowledge  In  manufacturing  responses  to  new  situations"  is  an  Import- 
ant type  of  learning  activity.  Its  occurrence  is  not  confined  to  solving 
verbal  problems. 

Thought  questions  that  do  not  Involve  arithmetical  calculations 
also  furnish  a  basis  for  using  knowledge.  Whenever  a  student  encount- 
ers a  difficulty  or  Is  asked  a  question  which  he  Is  unable  to  answer,  he 
has  a  problem  to  solve.  If  he  manufactures  a  solution  for  it  he  engages 
In  reflective  thinking  and  consequently  engages  in  learning  activity. 

6.  Tracing  the  thinking  of  another  person  by  listening  to  an  oral 
description  of  it  or  by  reading  a  printed  record  constitutes  a  sixth  type 
of  learning  activity.  It  occurs  In  arithmetic  when  a  pupil  listens  to  an 
explanation  of  the  solution  of  a  problem  given  by  the  teacher  or  by  an- 
other pupil. 

7.  Expressing  one's  ideas  is  educative,  particularly  when  attention 
Is  given  to  their  evaluation  and  organization.  A  pupil  learns  by  ex- 
plaining the  solution  of  a  problem  but  the  amount  of  learning  may  be- 
come almost  negligible  if  he  merely  follows  a  definite  formula,  as  was 
frequently  required  in  the  teaching  of  mental  arithmetic.  Expression 
of  ideas  also  occurs  In  responding  to  thought  questions. 

"See  page  22. 

[29] 


8.  "Prolonging,  repeating   and  intensifying   one's  experiences,"® 

represents  a  type  of  learning  activity  which  is  very  prominent  in  arith- 
metic. "Drill"  or  '"practice"  clearly  comes  under  this  type  of  learning 
activity.  "Living  over"  perceptual  experiences,  recalling  what  has  been 
read  or  heard,  thinking  through  the  solution  of  a  problem,  retracing  an 
explanation  given  by  the  teacher,  reconnecting  a  meaning  with  an  ab- 
stract or  general  term,  and  the  like  are  also  illustrations  of  "repeating 
experiences." 

9.  Learning  activities  resulting  in  the  acquisition  of  general  pat- 
terns of  conduct.  A  description  of  the  activities  that  result  in  the  acquisi- 
tion of  general  patterns  of  conduct  is  difficult,  but  It  appears  that  the 
production  of  this  class  of  outcomes  tends  to  be  governed  by  subtle  fac- 
tors. The  specific  habits  necessary  for  responding  to  quantitative  rela- 
tionships in  which  one  number  is  missing"  can  be  engendered  by  having 
the  pupil  engage  in  appropriate  practice.  On  the  other  hand,  the  "habit 
of  accuracy''  does  not  result  from  engaging  in  any  certain  activities.  Two 
pupils  may  apparently  engage  in  the  same  learning  activities  and  one 
will  acquire  the  ''habit  of  accuracy"  while  the  other  will  not. 

General  patterns  of  conduct  have  been  described  as  by-products 
which  may  be  produced  in  the  acquiring  of  specific  habits  and  knowl- 
edge. This  appears  to  be  a  valid  description  but  it  should  be  noted 
that  the  statement  is  "may  be  produced"  rather  than  "are  produced." 
The  engendering  of  general  patterns  of  conduct  Is  Incidental  but  not 
accidental.  They  result  when  the  conditions  are  right;  they  do  not 
when  the  necessary  conditions  are  not  secured.  However,  our  knowl- 
edge of  this  phase  of  the  learning  process  is  not  yet  sufficient  for  us 
to  specify  in  detail  what  conditions  are  necessan.'  for  the  engendering 
of  a  general  pattern  of  conduci. 

Studying  and  reciting  frequently  involve  a  combination  of  types 
of  learning  activities.  The  preceding  analysis  of  learning  activity  Is  not 
intended  to  imply  that  each  type  occurs  separately  and  independently. 
Frequently  the  total  activity  of  the  pupil  Is  a  combination  of  two  or 
more  types.  Perhaps  a  more  accurate  statement  would  be  that  the 
learner,  in  doing  a  school  exercise,  may  shift  from  one  type  of  activity 
to  another.  For  example,  in  attempting  to  solve  a  verbal  problem  one 
may  "trace  the  thinking"  of  the  teacher  by  listening  to  an  explanation 


*This  phrase  is  used  by: 

BoBBiTT.  Fr.\n"klix.  How  to  Make  a  Curriculum.  New  York:  Houghton  Mifflin 
Company,  1924,  p.  57.  The  word  "experiences"  in  this  expression  has  a  somewhat 
broader  meaning  than  on  page  28. 

"See  page  18. 

[30] 


of  certain  phases  of  it  or  "comprehend  the  meaning  of  a  new  abstract 
term"  in  addition  to  thinking  reflectively  about  the  problem. 

Relation  between  types  of  learning  activity  and  rubrics  of  abili- 
ties. In  the  preceding  pages  we  have  described  the  types  of  learning 
activity  in  which  children  engage  and  the  types  of  abilities  that  are  pro- 
duced as  outcomes.  Perhaps  the  reader  has  already  raised  the  ques- 
tion: just  how  are  the  different  types  of  activity  related  to  the  different 
kinds  of  ability;  or  more  specifically,  if  a  pupil  engages  in  a  specified 
kind  of  learning  activity,  what  will  be  the  nature  of  the  resulting  out- 
comes. 

Although  the  relation  between  mental  activity  and  the  resulting 
outcomes  cannot  be  stated  in  terms  of  precise  laws,  such  as  have  been 
formulated  in  chemistry  and  physics,  it  is  possible  to  state  certain  gen- 
eral laws.  Any  ability  (specific  habit,  knowledge,  or  general  pattern  of 
conduct)  may  be  thought  of  as  a  response  connected  with  a  stimulus 
and  its  quality  depends  upon  the  strength  of  the  connection  as  well  as 
upon  the  response.  For  example,  6x7=  is  a  stimulus,  and  42  the 
response.  The  strength  of  the  connection  between  6x7=  and  42  is  a 
very  important  element  in  a  pupil's  ability  to  respond  to  6  X  7. 

In  each  of  the  types  of  learning  activity  there  is  the  exercise  of 
connections  between  stimuli  (situation)  and  responses.  In  direct  ex- 
periencing the  connection  is  between  the  stimulus  apprehended  by 
means  of  a  sense  organ  and  resulting  percept.  In  "problem  solving" 
there  is  a  sequence  of  stimuli  and  responses,  both  of  which  usually  are 
ideas,  meanings,  concepts,  and  the  like.  In  solving  a  problem  a  pupil 
manufactures  a  "new  response"  (the  solution)  which  is  connected  with 
the  problem. 

Laws  governing  the  effectiveness  of  learning  activities.  The  effec- 
tiveness of  any  learning  activity  in  producing  mental  equipment  (spe- 
cific habits,  knowledge  or  general  patterns  of  conduct)  is  governed  by 
certain  laws'^  which  may  be  stated  as  follows: 

I.  If  other  factors  affecting  learning  remain  unchanged,  the 
strength  of  a  modifiable  connection  between  a  situation  and  a  response 
is  strengthened  as  it  is  exercised  and  up  to  a  certain  limit  the  strength 
of  the  connection  increases  with  the  amount  of  exercise  but  not  in  a 
constant  ratio. 


^These  "laws  of  learning"  are  based  on  formulations  by: 

Thorxdike,  E.  L.  Educational  Psychology,  Vol.  II.  Xew  York:  Teachers  Col- 
lege, Columbia  University,  1913.  452  p.,  or  Educational  Psychology,  Briefer  Course. 
New  York:    Teachers  College,  Columbia  University,  1914,  Part  II. 

Gates,  Arthur  I.  Psychology  for  Students  of  Education.  New  York:  The  Mac- 
millan  Company,  1925,  Chapter  X. 

[31] 


II.  Other  conditions  being  equal,  the  more  recent  the  exercise  of  a 
modifiable  connection  between  a  situation  and  a  response,  the  stronger 
the  connection  is.  This  implies  that  a  connection  which  is  not  exercised 
gradually  grows  weaker. 

III.  The  effect  of  the  exercise  of  a  modifiable  connection  between 
a  situation  and  a  response  depends  upon  the  degree  of  satisfaction  that 
accompanies  or  follows  the  activity.  Other  conditions  being  equal,  when 
"a  satisfying  state  of  affairs"  prevails  the  connection  is  strengthened; 
when  a  state  of  dissatisfaction  or  annoyance  prevails,  the  connection  is 
weakened. 

IV.  The  strengthening  effect  of  the  exercise  of  a  modifiable  con- 
nection between  a  situation  and  a  response  depends  upon  the  distribu- 
tion of  the  exercise,  and  other  things  being  equal  the  maximum  effect 
is  obtained  by  distributing  the  exercise  rather  than  by  concentrating  it. 

\'.  The  subject's  capacity  to  learn  (commonly  called  general  intel- 
ligence) contributes  to  the  effect  of  the  exercise  of  a  modifiable  connec- 
tion between  a  situation  and  a  response. 

Predicting  activity  necessary  for  attainment  of  specific  objectives. 
When  the  teacher  has  formulated  her  immediate  objectives,  that  is,  cer- 
tain specific  habits,  items  of  knowledge  or  general  patterns  of  con- 
duct to  be  engendered,  she  then  must  predict  the  learning  activities  in 
which  it  will  be  necessary  for  her  pupils  to  engage  in  order  to  attain 
these  objectives.  For  example,  suppose  the  immediate  objectives  are 
the  fixed  associations  designated  as  the  multiplication  combinations,  the 
teacher's  problem  is:  'Tn  what  activities  must  I  get  my  pupils  to  en- 
gage in  order  to  learn  these  fixed  associations?"  Our  knowledge  of  the 
relations  between  learning  activity  and  outcomes  enables  us  to  con- 
clude that  perceptual  experiencing  is  necessary  to  provide  a  founda- 
tion for  the  concept  of  multiplication.  There  must  also  be  generalizing 
of  these  experiences  and  of  course  many  repetitions  of  the  exercise  of 
the  connections  between  the  products  and  the  numbers  whose  products 
are  beinR  learned. 


[32] 


CHAPTER  III 
THE  LEARNING  EXERCISES  OF  ARITHMETIC 

The  problem  of  this  chapter.  It  is  the  problem  of  this  chapter  to 
identify  and  describe  the  kinds  of  exercises  which  may  be  assigned  by 
teachers  as  the  basis  for  the  activity  necessary  for  the  attainment 
of  the  objectives  described  in  Chapter  I.  No  attempt  will  be  made  to 
determine  the  relative  effectiveness  of  the  several  types  of  exercises  or  to 
indicate  when  each  should  be  used. 

Classes  of  learning  exercises.  The  analysis  of  learning  activity  in 
the  preceding  chapter  might  be  used  as  a  basis  for  a  corresponding  list 
of  types  of  exercises  but  it  has  seemed  desirable  to  recognize  other  fac- 
tors than  the  general  type  of  ensuing  mental  activity.  For  example, 
counting  objects  and  measuring  linear  distance  involve  perceptual  ex- 
periencing but  they  are  sufficiently  different  to  justify  listing  them  as 
two  separate  types  of  exercises.  The  following  list  is  not  offered  as  a 
complete  enumeration  of  the  classes  of  requests^  that  teachers  of  arith- 
metic make  of  their  pupils  but  it  will  serve  to  indicate  their  range. 

1.  Requests  to  count  objects  such  as  the  children  in  the  class, 
the  windows  in  the  classroom,  marks  on  the  blackboard  or  in  the 
textbook,  and  the  like. 

2.  Requests  to  measure  magnitudes  such  as  length  of  desk  or 
room,  the  water  in  a  pail,  and  the  like. 

3.  Requests  to  estimate  physical  magnitudes. 

4.  Projects  and  construction  exercises  including  those  involv- 
ing cooking,  sewing,  gardening,  and  the  like.  These  involve  implied 
requests  to  detect  and  formulate  needs  for  measuring,  computing, 
keeping  records,  and  the  like. 

5.  Games  involving  quantitative  activities  such  as  keeping 
score. 

6.  Requests  to  visit  such  places  as  a  retail  store  or  bank  in 
order  to  observe  adult  activities. 

7.  Requests  to  describe  perceptual  experiences  relating  to 
arithmetic. 


^An  analysis  of  two  of  the  classes,   examples   and  verbal   problems,   is  given,   be- 
ginning p.  35. 

[33] 


8.  Accounts  of  experiences  or  other  descriptions  to  be  listened 
to  or  read. 

9.  Requests  to  generalize  experiences.  (Usually  these  requests 
are  not  direct.   See  page  28.) 

10.  Dramatization  of  an  adult  activity  such  as  farming,  man- 
ufacturing, banking,  or  keeping  store. 

11.  Dictation  exercises  in  which  numbers  and  other  arithmet- 
ical symbols  are  to  be  written. 

12.  Requests  to  copy  numbers  and  other  arithmetical  symbols 
from  the  text  or  blackboard. 

13.  Requests  to  read  orally  numbers  and  other  arithmetical 
symbols. 

14.  Requests  to  repeat  orally  or  to  write  certain  groups  of 
symbols  or  facts.  This  includes  counting  by  2's,  3's,  and  so  forth 
as  well  as  groups  commonly  designated  as  "tables." 

15.  Requests  to  memorize,  that  is,  to  repeat  lists  of  facts,  tech- 
nical terms,  abbreviations  and  the  like  without  specific  exercises 
calling  for  repetitions. 

16.  Requests,  both  oral  and  written,  for  the  missing  number 
in  a  specific  quantitative  relationship,  such  as  7  -(-  5  =  ?, 
27  +  6  =  .^  24  ^  4  =  .',  47  ^  9  =  ?,  .1214  =  ?,  1  yd.  =  r  ft. 

17.  Explicit  requests  to  perform  specified  calculations,  com- 
monly called  "examples."-    (For  types  of  examples  see  page  35.) 

18.  Verbal  problems.''  (For  types  of  verbal  problems  see  page 
41.) 

19.  Requests  to  explain  performed  calculations  or  solutions  of 
problems. 

20.  Requests  to  read  or  listen  to  an  explanation. 

21.  Requests  to  check  calculations. 

22.  Requests  to  inspect  and  verify  solutions  of  problems. 

23.  Fact  questions  other  than  requests  to  supply  the  missing 
number  in  a  quantitative  relationship.  (Questions  concerning 
functional  relationships  would  be  included  here  when  the  pupil  does 
not  find  It  necessary  to  think  reflectively  in  answering  them.) 


'For  definition  of  example  see  page  19. 

^Usually  a  verbal  problem  is  to  be  solved  but  a  pupil  may  be  requested  to  esti- 
mate the  answer. 

[34] 


24.  Thought  questions.*     (These  include  questions  concerning 
general  quantitative  relations  and  problems  without  numbers.) 

25.  Requests  to  read  and  comprehend  descriptions,  definitions, 
rules  and  abstract  terms." 

26.  Requests  to  read,   or   reproduce,   business   forms   such   as 
checks,  notes,  sales  slips,  and  so  forth. 

27.  Requests  to  collect  or  formulate  problems. 

28.  Requests  to  collect  quantitative  information  such  as  prices 
or  other  items  in  regard  to  business  practices. 

29.  Graphs  to  be  read. 

30.  Groups  of  data  to  be  represented  graphically. 

31.  Requests  to  use  tables  and  other  calculating  devices. 
Variations  within  the  classes  of  learning  exercises.    Each  of  the 

classes  in  the  preceding  list  includes  learning  exercises  that  differ  in 
certain  respects.  Some  of  these  differences  are  significant  but  others 
are  not.  The  exercises  included  in  the  first  class,  "Requests  to  count 
objects,"  differ  with  respect  to  the  kind  of  objects  to  be  counted.  Such 
a  difference  has  little  or  no  significance  because  the  counting  of  objects 
is  essentially  the  same  as  a  learning  activity  regardless  of  the  nature  of 
the  objects  counted.^'  On  the  other  hand,  in  the  second  class  the  meas- 
urement of  linear  distance  differs  in  a  significant  way  from  the  meas- 
urement of  mass  because  the  instruments  and  units  are  different. 

"Specific  requests  to  perform  certain  calculations"  (examples)  and 
"verbal  problems"  represent  very  complex  classes  of  learning  exercises. 
Since  "examples"  and  "problems"  are  used  extensively  as  bases  of 
learning  activity  in  the  field  of  arithmetic,  it  will  be  helpful  to  note 
the  types  of  exercises  included  in  each  of  these  classes. 

Types  of  examples.  The  term  "example"  is  used  as  the  name  for 
an  explicit  request  to  add,  subtract,  multiply,  or  divide,"  the  numbers 
being  given.  The  example  may  call  for  two  or  more  of  these  opera- 
tions to  be  performed  but  in  all  cases  the  request  is  explicit.  The  re- 
quest for  the  calculation  may  be  expressed  in  terms  of  symbols,  such 


■'For  a  general  discussion  of  types  of  thought  questions,  see: 

Monroe,  Walter  S.,  and  Carter,  Ralph  E.  "The  use  of  different  types  of 
thought  questions  in  secondary  schools  and  their  relative  difficulty  for  students."  Uni- 
versity of  Illinois  Bulletin,  Vol.  20,  No.  34.  Bureau  of  Educational  Research  Bulletin 
No.  14.   Urbana:    University  of  Illinois,  1923.    26  p. 

^Formulae  may  be  added. 

This  statement  is  not  intended  to  imply  that  a  pupil's  activity  is  essentially  the 
same  in  all  cases.  The  point  made  is  that,  other  things  being  equal,  the  nature  of  the 
objects  counted  is  not  significant. 

'The  extraction  of  roots  may  be  added  as  a  fifth  calculation   process. 

[35] 


as  694 -(- 27  ^  r,  37)848,  or  technical  terms  may  be  used,  as  ''Find 
the  product  of  87  and  64,"  "Divide  694  by  27." 

When  considered  as  learning  exercises,  it  is  obvious  that  examples 
which  differ  in  any  way  do  not  afford  the  basis  for  identical  mental 
activities.  The  connections  exercised  by  responding  to  646  X  23  are 
different  from  those  exercised  by  responding  to  646  X  67.  However,  the 
difference  is  dissimilar  to  that  existing  between  the  responses  to  "sub- 
tract 746  from  9286"  and  "divide  18%  by  P/f"  In  the  case  of  the  re- 
sponses to  646  X  ^^  snd  646  X  67,  we  may  say  that  they  are  similar 
in  the  sense  that  each  involves  the  exercises  of  multiplication  and  addi- 
tion combinations.  This  condition  of  similarity  is  expressed  by  saying 
all  examples  calling  for  the  multiplication  of  a  three-place  integer  by  a 
two-place  multiplier  constitute  a  type.  Examples  such  as  387  X  6  pro- 
vide a  sufficiently  different  learning  activity  to  justify  recognition  as 
another  type.  A  request  to  multiply  412  X  4  constitutes  a  third  type 
since  no  carrying  is  involved. 

Recognition  of  differences  of  the  kind  illustrated  in  the  preceding 
paragraph  raises  the  the  question,  "\\'hat  are  the  significant  types  of 
arithmetical  examples:"  Those  involving  only  one  calculation  process 
fall  naturally  into  four  general  groups;  (1)  addition,  (2)  subtraction, 
(3)  multiplication,  and  (4)  division.  Within  each  of  these  groups  three 
subdivisions  are  created  by  the  three  types  of  numbers;  integers,  com- 
mon fractions  and  decimals.  Each  of  these  twelve  divisions  obviously 
includes  examples  that  differ  in  certain  respects.  Some  of  these  differ- 
ences have  been  recognized  in  arithmetic  texts  for  many  years  by  ex- 
plicit "cases"  such  as  "short  multiplication."  "subtraction  with  borrow- 
ing" and  the  like.  However,  the  "cases"  usually  mentioned  do  not 
appear  to  constitute  a  complete  enumeration  of  the  types  of  examples. 
In  addition  a  long  column  of  figures  (12  to  15)  appears  to  constitute  a 
different  type  of  example  from  that  furnished  by  a  short  column  of  fig- 
ures (3  to  5).  In  multiplication  the  presence  of  zeros  in  the  multiplier 
appears  to  create  at  least  one  separate  type  of  example  and  possibly  two. 

We  have  little  experimental  evidence  concerning  the  differences  that 
must  exist  between  two  examples  in  order  to  require  that  they  be  listed 
as  belonging  to  separate  types.  The  following  lists  are  intended  to  be 
conservative.  Several  of  the  types  include  examples  which  differ  in 
certain  respects  and  it  may  be  that  the  differences  are  sufficiently  sig- 
nificant to  justify  the  recognition  of  subordinate  types.  However,  the 
enumeration  of  the  types  given  here  will  serve  to  show  the  general  char- 
acter of  the  learning  exercises  which  are  commonly  called  examples. 

[36] 


I.  ADDITION   OF   INTEGERS' 

1.  Short  column  addition,"  3 .  to  5  addends: 

8  3  9 
4  2  4 
7        4        S 


3 


2.  Long  column  addition.  (Two  or  more  sub-types  may  be  recognized  by  making 
divisions  on  basis  of  length.)  Frequently  columns  of  more  than  7  to  9  addends  con- 
stitute a  situation  different  from  that  furnished  by  an  example  of  5  to  7  addends  be- 
cause the  "span  of  attention"  is  increased  beyond  the  normal  length. 

3.  Addition  with  carrying. 

4.  Addition  of  numbers  of  different  lengths. 

II.  SUBTRACTION  OF  INTEGERS 

1.  Subtraction  of  a  number  of  one  digit  from  a  number  of  two  digits.  (Such 
subtractions  may  be  considered  as  additional  combinations  corresponding  to  the  ''higher 
decade  combinations"  in  addition.) 

2.  Subtraction  of  numbers  of  two  or  more  digits  involving  "borrowing,"  but  no 
zero  in  either  subtrahend  or  minuend. 

3.  Subtraction  of  numbers  of  two  or  more  digits  involving  "borrowing"  and  one 
or  more  zeros  in  the  minuend. 

840  507  1000  602 

73  184  63  276 

4.  Subtraction  of  numbers  of  three  or  more  digits  with  at  least  one  zero  in  the 
subtrahend. 

896  383 

170  207 

III.  MULTIPLICATION  OF  INTEGERS 

1.  Short  multiplication  with  carrying." 

2.  Long  multiplication  without  carrying. 

3.  Long  multiplication  with  carrying. 

4.  Multiplications  involving  one  or  more  zeros  in  multiplicand. 

8350  70S 

92  37 

5.  Multiplications  involving  one  or  more  zeros  in  multiplier. 

4736  845 

805  30 

IV.  DIVISION  OF  ONE  INTEGER  BY  ANOTHER 

1.  Short  division:   divisor  1  to  9,  no  zeros  in  quotient,  with  or  without  remainder. 


*An  example  may  be  expressed  in  two  or  more  wavs: 
18  18  +  33  +  187=     . 

33  Add  18,  33,  and  187. 

187  Find  the  sum  of  18,  33,  and   187. 

Such  variations  in  form  are  not  considered  here. 
"As  the  term  has  been  used  in  the  preceding  pages   (see  page  19)    the  basic  and 
secondary  combinations  do  not  constitute  examples. 

"Short  multiplication  without  carrying   might   be   listed   as   a    separate   type   but 
such  examples  are  essentially  only  groups  of  fundamental  combinations. 

[37] 


2.  Short  divisicn:  divisor  1  to  9,  one  or  more  zeros  in  quotient,  with  or  without 
remainder. 

3.  Long  division:^'  trial  quotient  true  quotient,  no  zeros  in  quotient,  no  carrying 
in  multiplications,  no  borrowing  in  subtractions  and  no  remainder.  (This  is  the  simplest 
type  of  long  division  example.) 

4.  Long  div'ision:  trial  quotient  true  quotient,  no  zeros  in  quotient,  and  no  re- 
mainder. (This  differs  from  the  preceding  by  permitting  carr>'ing  in  the  multiplications 
and  borrowing  in  the  subtractions.) 

5.  Long  division:  trial  quotient  true  quotient,  no  zeros  in  quotient  but  with 
remainder. 

6.  Long  division:  trial  quotient  not  true  quotient,  no  zeros  in  quotient  and  no 
remainder. 

7.  Long  division:  trial  quotient  not  true  quotient,  no  zeros  in  quotient,  but  with 
remainder. 

8.  Long  di\'ision:  zeros  in  quotient.  (This  may  be  considered  a  composite  of 
several  types.  WTien  a  zero  occupies  units  place  in  a  quotient  the  example  Is  probably 
different  from  those  in  which  it  appears  in  an  interior  position.  Differentiations  might 
also  be  made  in  respect  to  the  trial  quotient  and  the  remainder.) 

V.  ADDITION    OF   FRACTIONS'^ 

1.  Addition  of  two  or  more  fractions  with  common  denominators,  the  sum  being 
non-reducible,  that  is,  in  its  lowest  terms  and  less  than  unity. 

2.  Addition  of  two  or  more  fractions  with  common  denominators,  the  sum  being 
reducible. 

3.  Addition  of  two  or  more  fractions,  the  denominators  not  being  common.  (The 
examples  under  this  type  may  be  divided  according  to  the  reducible  quality  of  the  sum.) 

4.  Addition  of  mixed  numbers. 

5.  Addition  of  an  integer  and  a  fraction,  the  sum  to  be  expressed  as  an  improper 
fraction. 

VI.  SUBTRACTION  OF  FRACTIONS 
L  Subtraction  of  fractions  having  common  denominators.'^ 

2.  Subtraction  of  fractions  not  having  common  denominators. 

3.  Subtraction  of  a  fraction  from  a  mixed  number,  requiring  borrowing. 

4.  Subtraction  of  one  mixed  number  from  another,  not  requiring  borrowing. 

5.  Subtraction  of  one  mixed  number  from  another,  requiring  borrowing. 

6.  Subtraction  of  a  fraction  from  an  integer. 

VII.  MULTIPLICATION  OF  FRACTIONS 

1.  Multiplication  of  an  integer  greater  than  unitv  bv  a  unit  fraction  such  as  ^ 
or  1/5.  _ 

2.  Multiplication  of  an  integer  greater  than  unity  by  other  proper  fractions. 

3.  Multiplication  of  one  unit  fraction  by  another  unit  fraction. 

4.  Multiplication  of  two  fractions  neither  of  which  is  a  unit  fraction.  Product 
may  be  either  reducible  or  non-reducible. 


"The  possibility  of  a  large  number  of  types  of  long  division  examples  is  apparent 
from  the  fact  that  five  conditions  are  specified  in  defining  this  type.  The  list  given  here 
includes  only  what  appears  to  be  the  most  significant  types  of  long  division  examples. 

^For  a  much  more  elaborate  list  of  types  of  examples  In  the  addition  of  frac- 
tions see: 

Kallom,  Arthur  W'.  '"Analysis  of  and  testing  in  common  fractions,"  Journal  of 
Educational  Research,  1:177-92,  March.  1920. 

"The  character  of  the  difference  is  not  considered  a  differentiating  factor.  If  this 
were  done  additional  types  would  be  found. 

[38] 


5.  Multiplication  of  a  mixed  number  and  a  fraction.  Fractional  product  may 
be  reducible  or  non-reducible. 

6.  Multiplication  of  a  mixed  number  by  an  integer. 

7.  Multiplication  of  two  mixed  numbers. 

VIII.    DIVISION  OF  FRACTIONS" 

As  stated  on  page  36  the  list  of  types  presented  here  is  not  intended  to  include 
all  possible  ones  and  some  of  those  given  obviously  include  examples  which  differ  in 
certain  respects. 

1.  Division  of  an  integer  by  a  unit  fraction. 

2.  Division  of  an  integer  by  other  proper  fractions. 

3.  Division  of  one  fraction  by  another. 

4.  Division  of  a  fraction  by  an  integer. 

5.  Division  of  a  fraction  by  a  mixed  number.    ■ 

6.  Division  of  a  mixed  number  by  a  fraction. 

7.  Division  of  a  mixed  number  by  a  mixed  number. 

8.  Division  of  a  mixed  number  by  an  integer. 

9.  Division  of  an  integer  by  a  larger  integer. 

IX.  ADDITION  OF  DECIMALS" 

1.  Addends  form  addition  example  with  right  hand  margin  even  and  no  zeros  to 
the  left  of  the  last  significant  figure.^" 

.6      5.08  .4876 

.5      1.26  .8428 

.8      7.31  .9371 

.3     12.83  .8476 


2.  Addends  form  addition  example  with  right  hand  margin  uneven  but  no  zeros 
to  the  left  of  the  last  significant  figure. 

.6  17. 

.346  .8942 

.15  .327 

.3  1.25 


"It  would  be  possible  to  increase  the  number  of  example  types  under  this  and  the 
other  groups  by  listing  all  of  the  possible  combinations  of  the  conditions  affecting  the 
example.  In  the  case  of  the  division  of  fractions,  the  dividend  may  be  (1)  a  unit 
fraction,  (2)  other  proper  fractions,  (3)  an  improper  fraction,  (4)  a  mixed  number 
or  (5)  an  integer.  The  same  possibilities  exist  in  the  case  of  the  divisor.  The  quo- 
tient furnishes  another  basis  of  differentiation.  It  may  be  (1)  a  proper  fraction  in 
lowest  terms,  (2)  an  improper  fraction  in  lowest  terms,  (3)  a  proper  fraction  but  not 
in  lowest  terms,  (4)  an  improper  fraction  not  in  lowest  terms,  or  (5)  an  integer.  An 
indication  of  the  number  of  possible  types  of  examples  under  division  of  fractions  is 
given  In  an  article  by: 

Knight,  F.  B.  "A  note  on  the  organization  of  drill  work,"  Journal  of  Educa- 
tional Psychology,  16:108-13.  February,  1925.  In  this  article  the  division  of  fractions 
is  divided  into  55  units  of  skill. 

*The  significant  differences  between  addition  of  integers  and  addition  of  deci- 
mals have  not  been  determined.  It  appears  to  be  a  reasonable  hypothesis  that  the 
differences  are  confined  to  (1)  placement  of  the  decimal  point  in  the  sum,  (2)  the 
possible  uneven  right  hand  side  of  the  addition  example,  and  (3)  the  possible  presence 
of  zeros  between  the  decimal  point  and  the  first  significant  figure  of  the  addends.  It 
does  not  seem  that  the  presence  of  integers  in  the  addends  either  separately  or  in 
combination  with  a  decimal  should  constitute  a  significant  characteristic.  A  similar 
statement  may  be  made  in  the  case  of  subtraction. 

^*The  "last"  figure  is  the  one  farthest  left. 

[39] 


3.  Addends  have  zeros  to  the  left  of  the  last  significant  figure. 

.05  .05 

.0082  .06 

.075  .04 

.04  .03 


X.  SUBTRACTION  OF  DECIMALS 

1.  The  right  hand  figure  of  the  subtrahend  is  written  under  the  right  hand  figure 
of  the  minuend  and  no  zeros  to  the  left  of  the  last  significant  figure  in  either  decimal. 

12.5  .75 

8.2  .42 

2.  The  right  hand  figure  of  the  subtrahend  farther  removed  from  the  decimal 
point  than  the  right  hand  figure  of  the  minuend,  but  no  zeros  to  the  left  of  the  last 
significant  figure  in  either  decimal. 

1.  .75 

.25  .125 

3.  The  right  figure  of  the  minuend  farther  removed  from  the  decimal  point 
than  the  right  hand  figure  of  the  subtrahend,  but  no  zeros  to  the  left  of  the  last  sig- 
nificant figure  In  either  decimal. 

.875 
.5 

4.  Zeros  to  the  left  of  the  last  significant  figure  in  at  least  one  of  the  decimals. 

.0025  .875 

.0042  .012 

XI.  MULTIPLICATION  OF  DECIMALS 

1.  Multiplication  of  a  decimal  by  an  Integer. 

.75            .875            1.25 
5  64 8 

2.  Multiplication  of  an  integer  by  a  decimal. 

845  950  837 

.06  .7  1.5 

3.  Multiplication  of  a  decimal  by  a  decimal. 

XII.  DIVISION  OF  DECIMALS'" 

1.  Division  of  a  decimal  by  an  Integer  with  no  remainder. 

2.  Division  of  a  decimal  by  an  Integer  with  a  remainder  that  may  be  completely 
expressed  by  additional  decimal  places  in  the  quotient. 

3.  Division  of  a  decimal  by  an  Integer  with  a  remainder,  quotient  to  be  carried  to 
a  specific  number  of  decimal  places. 

4.  Division  of  an  Integer  by  a  decimal.  (It  is  possible  that  sub-types  are  formed 
by  divisors  such  as  .6,  .06.  .0006,  1.06.) 

5.  Division  of  one  integer  by  another  with  a  remainder,  quotient  to  be  carried 
to  a  specific  number  of  decimal  places. 


"See: 

Monroe,  Walter  S.  "The  ability  to  place  the  decimal  point  In  division," 
Elementary  School  Journal,  18:287-93,  December,  1917.  This  investigation  Indicated 
that  many  pupils  do  not  place  the  decimal  point  in  division  by  applying  a  general 
rule  but  use  a  special  rule  or  device  for  difTerent  cases.  If  a  special  rule  or  device 
were  employed  for  all  possible  combinations  of  dividend  and  divisor,  the  number  of 
types  of  examples  would  be  large  even  If  the  quantities  were  restricted  to  relatively 
few  decimal  places.     Only  a  few  of  the  more  significant  types  are  given  here. 

[40] 


6.  Division  of  one  decimal  by  another  when  the  number  of  decimal  places  in  the 
dividend  equals  or  exceeds  those  of  the  divisor,  with  no  remainder.  ("Xo  remainder" 
implies  that  if  the  decimal  point  w'ere  removed  from  both  dividend  and  divisor,  the 
former  would  be  larger  than  the  latter.) 

7.  Same  as  the  preceding  except  with  a  remainder. 

8.  Division  of  one  decimal  by  another  when  the  number  of  decimal  places  In  the 
dividend  is  less  than  those  In  the  divisor. 

Types  of  problems.  When  we  examine  the  problem  Hsts  in  current 
arithmetics,  we  find  a  conspicuous  lack  of  uniformity  in  the  captions  by 
which  these  lists  are  designated  in  different  texts.  Formerly,  most  of 
the  problems  given  in  an  arithmetic  were  listed  under  such  captions 
as:  "Rule  of  three,  direct,"  "Rule  of  three.  Inverse,"  "Partnership," 
"Alligation,"  "Barter,"  "Practice,"  "Profit  and  Loss,"  "Trade  discount," 
"True  discount,"  "Partial  payments,"  "Exchange,"  and  so  forth. 
Changes  In  business  practices  and  in  the  activities  of  adults  apart  from 
the  carrying  on  of  business  have  created  new  "applications"  of  arithme- 
tic. Many  of  the  titles  formerly  used  as  captions  for  problem  lists  have 
been  discarded  and  new  ones  substituted.  The  result  Is  that  at  the 
present  time  we  have  no  generally  recognized  plan  for  classifying  the 
problems  of  arithmetic. 

The  buying  and  selling  of  commodities,  borrowing  money,  con- 
structing houses  and  other  buildings,  insuring  property,  carrying  on  a 
business,  and  the  like  create  many  arithmetical  problems.  This  sug- 
gests that  the  sources  of  problems  be  used  as  a  basis  for  their  classi- 
fication^^ but  such  a  plan  will  not  give  groups  which  approximate  homo- 
geneity with  respect  to  the  activity  required  in  solving  the  problems. 
Furthermore,  an  examination  of  the  problems  In  our  arithmetics  will 
reveal  a  number  of  problems  whose  source  is  not  easily  identified.  In 
some  cases  the  problem  does  not  appear  to  be  connected  with  any  par- 
ticular activity  or  the  suggested  adult  activity  might  be  changed  with- 
out affecting  the  problem.    The  following  are  typical: 

1.  "A  pail  of  milk  holding  2  gallons  is  to  be  poured  into  quart  bottles.  How 
many  bottles  will  be  needed?" 

2.  "Henry  caught  three  fish.  The  first  weighed  12  ounces,  the  second  10  ounces, 
and  the  third  15  ounces.     What  was  the  total  weight  of  the  three.'" 

3.  "On  a  vacation  trip  Robert  walked  6^  miles  the  first  day,  7  miles  the  second 
day,  and  5%  miles  the  third  day.     Find  the  total  distance  traveled." 

4.  "The  length  of  an  iron  rod  was  95i%6  inches.  After  It  was  heated  Its  length 
was  found  to  be  96%2  inches.     How  much  was  the  length  increased  by  heating.^" 


^"The  writer  has  employed  this   method  of  analysis.     See: 

Monroe,  W.-^lter  S.  "A  preliminary  report  of  an  Investigation  of  the  economy 
of  time  in  arithmetic."  Second  Report  of  the  Com.mittee  on  Minimum  Essentials  In 
Elementary  School  Subjects.  Sixteenth  Yearbook  of  the  National  Society  for  the  Study 
of  Education,  Part  I.  Bloomington,  Illinois:  Public  School  Publishing  Company,  1917, 
p.  111-27. 


[41] 


In  the  first  problem  "milk"  could  be  changed  to  "water,"  "syrup," 
or  any  other  liquid  without  changing  the  problem.  Furthermore,  "bot- 
tles" could  be  changed  to  "cans"  or  "jars."  Hence,  there  is  no  signifi- 
cant connection  between  the  problem  and  any  adult  activity.  A  similar 
conclusion  applies  to  the  other  problems. 

Thus  it  appears  that  there  are  two  general  classes  of  problems: 
A,  Operation  Problems,  those  not  identified  with  a  particular  activity 
or  identified  with  an  activity  that  does  introduce  a  technical  terminology 
peculiar  to  that  activity;  B,  Activity  Problems,  those  identified  with  a 
definite  activity  of  children  or  adults  which  introduces  a  technical  term- 
inology. 

Within  each  of  these  two  general  classes  of  problems, ^^  a  further 
difi"erentiation  may  be  based  upon  the  implied  question  concerning  the 
functional  relationship.  (See  page  19.)  All  problems  that  ask  the  same 
question  may  be  considered  to  form  a  problem  type  which  may  be  de- 
scribed by  designating  the  quantities  given  and  the  one  to  be  found. 
The  question  concerning  functional  relationship  is:  What  calculations 
are  to  be  performed  upon  the  given  quantities  in  order  to  obtain  the 
one  to  be  found.'' 

An  elaborate  study  of  the  problems  provided  by  texts-°  resulted  in 
the  identification  of  52  problem  types  in  the  field  of  "operation  prob- 
lems" and  281  problem  types  under  "activity  problems."-^  Descrip- 
tions of  representative  problem  types  are  given  in  the  following  pages. 
A  complete  list  of  all  problem  types  is  printed  in  Appendix  A.  See 
also  pages  90-92. 
Al  To  find  totals  by  addition,  given  two  or  more  items,  values,  etc. 
A3  To  find  the  amount,  or  number  needed,  by  multiplication,  given  a  magnitude  and 

the  number  of  times  it  is  to  be  taken. 


"*A  verbal  problem  in  arithmetic  is  a  description  of  a  quantitative  situation  or 
condition  plus  a  question  that  usually  requires  a  numerical  answer.  The  solving  of  this 
requires  the  determination  of  the  calculations  to  be  made  in  order  to  obtain  the  an- 
swer. The  basis  of  the  determination  of  the  calculations  to  be  performed  in  the  solv- 
ing of  a  problem  is  the  general  quantitative  relation  which  connects  the  quantities 
of  the  problem.  For  example,  consider  this  problem:  '"An  agent  sells  goods  on  a  com- 
mission of  10  percent.  How  much  does  he  remit  to  his  principal  for  sales  amounting 
to  $1150?" 

The  quantities  of  this  problem,  proceeds  (amount  remitted  to  principal),  rate  of 
commission  and  amount  of  sales  are  related  as  follows :  Proceeds  =  amount  of  sales  —  the 
product  of  amount  of  sales  and  rate  of  commission.  In  order  to  solve  this  problem 
rationally,  that  is,  by  reasoning,  it  is  necessary  that  one  answer  the  question,  "How  is 
the  amount  to  be  remitted  to  the  principal  calculated  from  the  amount  of  sales  and 
the  rate  of  commission?" 

^See  page  48  for  a  description  of  this  study.  The  "problem  types"  are  used  in 
explaining  the  process  of  solving  verbal  problems.    See  page  21. 

"A  few  tvpcs  of  learning  exercises  are  included  that  do  not  require  calculation. 
See  Appendix  A,  A24,  A25,  B5f,  B5h,  BSi,  B5j,  B5k,  BSl. 

[42] 


AS  To  find  how  many  times  a  stated  quantity  is  contained  in  a  given  magnitude,  given 

the  quantity  and  the  magnitude. 
A6  To  find  how  many  when  reduction  ascending  is  required,  given 

a.  a  magnitude  expressed  in  terms  of  a  single  denomination. 

b.  a  magnitude  expressed  in  terms  of  two  or  more  denominations. 
A8  To  find  a  dimension,  given  the  area  of  a  rectangle  and  one  side. 

A13  To  find  a  difference,  given  denominate  numbers  of  different  denominations. 
A15  To  find  the  ratio  of  one  number  to  another,  given  the  two  numbers. 
A16  To  find  a  part  of  a  number,  given  the  ratio  of  the  part  to  the  number  and  the 
number.     (The  fraction  may  be  in  terms  of  fractions  or  decimals.) 

Bl  Buying  and  selling, ^^  simple  cases.  1 

a.  To  find  the  total  price:" 

1.  given  the  number  of  units  and  price^^  per  unit. 

2.  given  the  number  of  units  and  the  price  per  unit  of  another  denonination. 

b.  To  find  the  number  of  units: 

1.  given  the  total  price  and  price  per  unit. 

2.  given  the  total  price  and  the  price  per  unit  in  another  denomination. 

3.  received  in  exchange  of  commodities,  given  an  amount  of  each  commodity 
and  the  unit  for  each. 

4.  given  the  price  per  unit  of  each  of  two  commodities,  the  total  price  of  both, 
and  the  ratio  of  the  number  of  units  of  one  to  the  number  of  units  of  the 
other.  \ 

5.  given  the  margin^^  per  unit  and  the  total  margin.  \ 

c.  To  find  the  price  per  unit:  \ 

1.  given  the  total  price  and  the  number  of  units. 

2.  given  the  total  price  and  the  number  of  units  in  another  denomination. 

3.  in  exchange  of  commodities,  given  the  number  of  units  of  each  commodty 
and  the  price  per  unit  of  one.  \ 

4.  given  the  number  of  units  of  each,  the  combined  price  of  both,  and  the  ratio 
of  the  price  of  the  one  to  that  of  the  other.  \ 

d.  To  find  the  amount  to  be  received  for  several  items,  given  the  price  of  each.  \ 

e.  To  make  change,  given  an  amount  of  money  and  the  price  of  a  commodity. 

f.  To  find  the  margin  or  loss  given  the  cost  price  and  the  selling  price. 

g.  To  find  the  total  margin  or  total  loss: 

1.  given  the  number  of  units  and  the  margin  or  loss  per  unit. 

2.  given  the  unit  cost,  the  unit  selling  price,  and  the  number  of  units. 

h.  To  find  the  margin  or  loss  per  unit,  given  the  total  margin  or  loss  and  the  num- 
ber of  units. 


^^Descriptions  of  quantitative  relations  given  below  are  expressed  in  terms  of  buy- 
ing. In  some  cases  changes  in  terminology  would  be  necessary  if  the  activity  were  to 
be  considered  from  the  standpoint  of  selling. 

^'"Total  price"  is  used  to  designate  the  amount  received  for  several  units  of  the 
same  commodity  rather  than  the  amount  received  for  several  commodities. 

"Price  is  used  to  designate  the  quantity  taken  as  a  basis  of  computation.  Usu- 
ally "price"  refers  to  the  value  or  worth  of  a  unit  rather  than  a  specified  number  of 
units.     "Price"  is  often  limited  by  the  qualifying  terms  cost,   selling,  marked,   and  list. 

""Margin  is  a  term  used  to  represent  the  difference  between  the  cost  price  and  the 
selling  price  and  therefore  is  a  substitute  for  the  words  "gain"  and  "profit"  as  they 
are  commonly  used. 


[43] 


B2  Buying  and  selling,  more  complex  types. 

a.  To  find  the  selling  price: 

1.  given  the  rate"°  of  discount  or  loss,  and  the  price. 

2.  given  the  rate  of  advance  or  margin  and  the  price. 

3.  given  the  rate  of  two  or  more  successive  discounts  and  the  price. 

4.  given  the  price,  rate  of  advance  or  margin,  and  rate  of  discount  or  loss. 

5.  given  the  rate  of  commission,  discount,  margin,  or  loss  and  the  amount  of 
commission,  discount,  margin,  or  loss. 

6.  given  the  price  and  the  amount  of  commission  or  discount. 

b.  To  find  the  amount  of  margin,  loss,  commission,  or  discount: 

i.  given  the  total  price  and  the  rate  of  margin,  loss,  commission,  or  discount. 

2.  given  two  or  more  successive  discounts  and  the  total  price. 

3.  given  the  total  price  and  the  selling  price. 

c.  To  find  the  rate  of  margin,  loss,  discount,  advance,  or  commission: 

1.  given  the  total  price  and  the  amount  of  margin,  loss,  discount,  advance,  or 
commission. 

2.  given  the  cost  price  in  terms  of  two  successive  rates  of  discounts  and  the  list 
price,  and  the  selling  price  in  terms  of  a  single  rate  of  discount  and  the  list 
price. 

3.  given  the  total  price  and  the  selling  price. 

d.  To  find  the  price: 

1.  given  the  selling  price  and  the  rate  of  discount  or  loss. 

2.  given  the  amount  of  margin,  loss,  commission,  or  discount  and  the  rate  of 
margin,  loss,  commission,  or  discount. 

3.  given  the  selling  price  and  rate  of  margin. 

4.  given  the  selling  price  and  two  or  more  successive  discounts. 

e.  To  find  the  amount  due  the  agent  or  agents,  given  the  number  of  units,  the 
price  per  unit,  and  the  rate  of  commission. 

f.  To  find  the  equivalent  single  discount  in  percent,  given  two  or  more  successive 

rates  of  discount, 
g.  To  find  one  of  two  or  more  successive  discounts,  given  the  list  price,  one  or  more 

of  the  successive  discounts  in  percent,  and  the  net  price. 

Limitations  of  the  list  of  problem  types.  A  comparison  of  the 
problem  types  appearing  under  the  caption,  operation  problems,  with 
those  listed  as  activity  problems  reveals  a  number  of  apparent  dupli- 
cations. This  is  to  be  expected  because  it  is  theoretically  possible  for 
any  question  concerning  a  quantitative  relationship  listed  under  oper- 
ation problems  to  be  implied  in  a  problem  clearly  identified  with  some 
activity.  When  this  occurs  the  problem  has  been  classified  as  belonging 
to  a  type  under  activity  problems. 

The  recognition  of  two  overlapping  groups  of  problems  appeared 
to  be  justified  by  the  fact  that  many  problems  found  in  arithmetic 
texts  could  not  be  assigned  to  an  activity  and  that  when  they  were 
clearly  identified  with  a  particular  activity  such  as  "borrowing,  lending 
or  saving  money"  or  "insurance"'  a  technical  terminology  was  intro- 
duced which  tended  to  make  them  different  from  other  problems  re- 
quiring the  same  calculations  but  not  identified  with  the  same  activity. 


"Rate  may  be  expressed  in  terms  of  percent  or  as  a  fraction. 

[44] 


Failure  to  group  problem  types  under  some  such  heading  as  "activity 
problems"  would  suggest  that  the  problems  of  arithmetic  were  abstract. 
The  absence  of  a  list  of  "operation  problems"  would  have  made  it  im- 
possible to  classify  many  problems  now  found  in  arithmetic  texts. 

A  comparison  of  certain  of  the  groups  of  problem  types  under  act- 
ivity problems  (e.g.,  Bl,  B2  and  B3)  will  reveal  a  type  of  duplication 
caused  by  the  fact  that  two  bases  of  differentiation  were  recognized; 
first,  the  general  character  of  the  activity  in  which  problems  occur,  and 
second,  the  question  concerning  functional  relations  which  a  problem 
implies.  It  was  decided  that  the  first  basis  (general  character  of  the 
activity)   should  have  precedence  over  the  second. 

The  writer  and  his  assistants  were  compelled  to  exercise  judgment 
on  a  number  of  other  points.  Consequently,  the  list  of  problem  types 
should  not  be  accepted  as  final.  Especially,  the  conclusion  that  there 
are  exactly  333  problem  types  should  not  be  drawn.  This  total  would 
have  been  different  if  different  decisions  on  a  number  of  minor  points 
had  been  made. 

Value  of  the  list  of  problem  types.  Although  the  list  of  problem 
types  has  been  evolved  after  much  careful  thought  and  has  been  used 
as  a  basis  in  analyzing  ten  series  of  arithmetic  texts,  the  enumeration 
of  problem  types  given  in  Appendix  A  must  be  considered  only  a  ten- 
tative formulation  representing  the  judgment  of  the  writer  and  his  as- 
sistants. However,  this  tentative  formulation  should  prove  useful  be- 
cause it  emphasizes  that  an  arithmetical  problem  asks  a  question  con- 
cerning a  quantitative  relationship  which  the  solver  of  the  problem 
must  identify  and  then  answer.  Furthermore,  it  provides  a  workmg 
basis  for  considering  the  problem  content  of  arithmetics. 

Conclusions  in  regard  to  learning  exercises.  The  most  significant 
conclusion  to  be  drawn  from  this  description  of  the  learning  exercises 
of  arithmetic,  especially  the  types  of  examples  and  problems,  is  that  the  ^_^-— 
number  of  types  of  exercises  is  large.  Each  type  of  exercise  constitutes 
a  basis  for  a  learning  activity  which  is  different,  at  least  in  some  re- 
spects, from  that  occurring  in  a  pupil's  response  to  any  other  type. 
Hence,  this  analysis  of  the  learning  exercises  of  arithmetic  is  a  neces- 
sary prerequisite  for  a  consideration  of  the  teacher's  responsibility  for  ^y 
devisine  and  selecting  exercises. 


[-^5] 


CHAPTER  IV 

THE  LEARNING  EXERCISES  PROVIDED  BY  TEXTS 
IN  ARITHMETIC 

Three  types  of  content  in  arithmetic  texts.  Arithmetic  texts  in- 
clude three  general  types  of  content;  (1)  statements  of  what  pupils  are 
to  learn  (facts,  rules,  definitions,  and  principles);  (2)  illustrations,  ex- 
planations, descriptions,  and  the  like  which  imply  learning  exercises  in- 
volving tracing  (see  numbers  8,  20,  and  26,  page  34);  and  (3)  explicit 
learning  exercises.  Examples  (explicit  requests  to  perform  specified 
calculations)  and  verbal  problems  make  up  the  majority  of  this  third 
type  of  material. 

The  problem  of  this  chapter.  The  problem  of  this  chapter  is  to 
present  certain  information  relative  to  the  example  and  problem  con- 
J  tent  of  arithmetic  texts.  The  information  relating  to  provisions  for  the 
first  type  of  learning  exercise  is  taken  from  studies  reported  by  other 
investigators.  The  mformation  concerning  the  problem  content  is  based 
on  an  original  investigation  conducted  under  the  direction  of  the  writer. 

The  example  content  of  arithmetic  texts.  Since  the  principal 
function  of  examples  is  to  provide  practice  on  the  combinations  (basic 
and  secondary),  the  most  significant  information  relative  to  the  example 
content  of  arithmetic  texts  is  the  number  of  occurrences  of  each  of  the 
combinations.  A  statement  of  the  amount  of  space  devoted  to  ex- 
amples or  the  number  of  learning  exercises  of  this  type  does  not  con- 
stitute a  very  significant  description.^  An  analysis  of  the  examples  with 
respect  to  the  operations  involved  is  a  little  more  helpful  but  it  still 
leaves  one  with  only  a  very  vague  notion  of  the  nature  of  the  learning 
exercises  which  the  texts  provide. 

Writers-   who  have   analyzed   the  example  content^  of   arithmetic 


'For  an  illustration  of  an  analysis  of  this  type  see: 

Spaulding,  F.  T.  "An  analysis  of  the  content  of  six  third-grade  arithmetics," 
Journal  of  Educational  Research,  4:415-23,  December,  1921.  The  investigator  presents 
a  count  of  the  examples  and  problems  in  six  third-grade  arithmetics.  He  found  that 
the  ratio  of  examples  to  problems  varied  from  nearly  5  to  1  to  approximately  2  to  1, 
the  average  being  a  little  more  than  3  to  1. 

■Clapp,  Frank  L.  'The  number  combinations:  their  relative  difficulty  and  the 
frequency  of  their  appearance  in  textbooks."  Bureau  of  Educational  Research  Bulletin 
No.  2.    Madison,  Wisconsin:  University  of  Wisconsin,  July,  1924.     126  p. 

Knight,  F.  B.  "A  note  on  the  organization  of  drill  work,"  Journal  of  Educa- 
tional Psychology,  16:108-17,  February,  1925. 

[46] 


texts  agree  that  the  provisions  for  practice  on  the  different  combinations 
vary  greatly,  and  that  pupils  who  do  all  of  the  examples  provided  by  a 
given  text  will  receive  more  practice  on  the  easier  combinations  than  on 
the  more  difficult  ones.*  For  example.  Clapp^  reports  the  following  fre- 
quencies of  combinations  In  Book  II  of  a  certain  series  of  arithmetics: 
1  +  1  =  ,434  times;  2  +  1  =  ,444  times;  1  +  2  =  ,299  times; 
4+1=  ,447  times;  7  +  5=  ,76  times;  7  +  6=  ,74  times; 
8  +  7=  ,  102  times;  6  +  8=  ,  62  times;  7  +  9  =  ,  74  times.  Os- 
burn  reports  that  ''one  hundred  and  eighty  out  of  a  total  of  1,325  com- 
binations do  not  occur  at  all  in  the  book  considered*^  while  some  easy 
ones  occur  more  than  300  times."  Clapp  reports  the  following  coeffi- 
cients of  correlation  between  the  difficulty  of  combinations  and  the  fre- 
quency of  their  appearance  in  textbooks:  addition  — .452  ±  .054; 
subtraction  —.329  ±.061;  multiplication  — .384  ±  .057;  division 
—  .421  ±1  .061.  These  results  are  for  Text  A.  Similar  coefficients  of 
correlation  are  given  for  Text  B.  Since  all  of  the  coefficients  are  nega- 
tive and  "large"  in  comparison  with  the  probable  error,  they  mean  the 
more  difficult  the  combination  the  less  frequently  it  occurs." 

As  might  be  expected,  when  texts  are  compared  with  reference  to 
their  provisions  for  practice  on  the  combinations,  there  Is  a  conspicu- 
ous lack  of  similarity  In  their  example  content.  With  the  possible  ex- 
ception of  texts  published  since  the  results  of  the  first  analyses  have 
been  available.  It  appears  certain  that  the  practice  a  pupil  receives  upon 
the  combinations  of  arithmetic,  both  basic  and  secondary,  will  not  be 
adjusted  to  the  difficulty  of  the  combinations,  and  that  the  amount  of 
practice  upon  the  different  combinations  will  depend  upon  the  text  he 
studies. 


Thorndike,  Edward  L.  The  Psychology  of  Arithmetic.  New  York:  The  Mac- 
millan  Company,  1922,  Chapter  VL 

^■"Example  content"  Includes  both  examples  as  defined  on  page  19  and  requests 
for  the  fundamental  combinations. 

*Since  "easy"  combinations  are  those  which  pupils  respond  to  with  the  fewest 
errors  and  the  "difficult"  combinations  are  those  which  pupils  know  least  well,  one  might 
Insist  that  the  combinations  found  to  be  "easy"  possessed  this  quality  because  the  texts 
provided  much  drill  on  them  and  that  the  "difficult"  ones  were  not  known  so  well  be- 
cause the  pupils  were  not  given  as  much  opportunity  to  learn  them.  However,  a  care- 
ful study  of  the  available  data  does  not  support  this  hypothesis.  It  appears  that  cer- 
tain combinations  are  inherently  more  difficult  than  others. 

'Clapp,  Frank  L.  op.  at. 

*This  is  described  as  Book  I  of  a  widely  used  series  of  arithmetics. 

'An  analysis  of  the  practice  exercises  prepared  by  Courtis  and  by  Studebaker  re- 
veals similar  conditions.     See: 

OsBURN,  W.  J.  "A  study  of  the  validity  of  the  Courtis  and  Studebaker  Practice 
Tests  In  the  Fundamentals  of  Arithmetic,"  Journal  of  Educational  Research,  8:93-105, 
September,  1923. 

[471 


The  distribution  of  practice.  In  considering  the  learning  exercises 
provided  by  a  series  of  arithmetics,  it  is  important  to  note  the  distribu- 
tion of  practice  as  well  as  the  nature  of  this  practice.  Thorndike*  has 
shown  that,  in  certain  texts  which  are  probably  representative,  the  prac- 
tice is  distributed  in  a  way  that  appears  to  represent  inefficient  instruc- 
tion. Investigations  in  the  psychology  of  learning  indicate  that  in 
learning  the  combinations  of  arithmetic  there  should  be  a  reasonably 
large  number  of  repetitions  during  the  first  learning  period  and  a  grad- 
ual decrease  in  their  number  during  subsequent  periods  which  should 
occur  at  gradually  increasing  intervals.  Thorndike  found  that  the 
amount  of  practice  on  5  X  5  in  the  first  two  books  of  a  three-book 
series  increased  as  the  pupil  advanced  through  the  series.  He  suggests 
that  the  distribution  of  practice  in  this  combination  ''would  be  better 
if  the  pupil  began  at  the  end  and  went  backwards." 

Problem  content  of  arithmetic  texts.  In  order  to  determine  the 
nature  of  the  problem  content^  of  arithmetic  texts,  the  list  of  333  prob- 
lem types  described  in  Chapter  III  was  used  as  a  basis  for  analyzing 
the  second  and  third  books  of  ten  three-book  series  of  arithmetics.^" 
Each  problem  in  these  books  was  read  and  a  decision  made  in  regard 
to  the  question  it  Implicitly  asked  concerning  a  functional  relationship.^^ 


*Thorndike,  Edward  L.  The  Psychology  of  Arithmetic.  New  York:  The  Mac- 
millan  Company,  1922,  Chapter  \TII. 

^"Problem  content"  does  not  include  explicit  requests  for  a  definite  calculation, 
such  as  'What  is  7  percent  of  ^7400.00.'"  or  '"Reduce  2  miles  to  yards."  Such  exer- 
cises were  considered  examples. 

"The  series  analyzed  are: 

Anderson,  Robert  F.  The  Anderson  Arithmetic.  New  York:  Silver,  Burdett  and 
Company,  1924. 

Alexander,  Georgia,  and  Dewey,  John.  The  .Alexander-Dewey  .Arithmetic.  New 
York:  Longmans,  Green  and  Company,  1921. 

Drushel,  J.  Andrew,  Noonan,  Margaret  E.,  and  Withers,  John  W.  .Arith- 
metical Essentials.     New  York:  Lyons  and  Carnahan.  1921. 

Hamilton,  Samuel.  Hamilton's  Essentials  of  Arithmetic.  Higher  Grades.  New 
York:  American  Book  Company.  1920. 

HoYT,  Franklin  S.,  and  Peet,  Harriet  E.  Ever^'day  .Arithmetic.  New  York: 
Houghton  MifBin  Company,  1920. 

Lennes,  N.  G..  and  Jenkins,  Frances.  Applied  .Arithmetic.  Tlie  Tliree  Essen- 
tials.    Philadelphia:  J.  B.  Lippincott  Company,  1920. 

Stone,  John  C,  and  Mii.lis,  J.^^mes  F.  New  Stone-Millis  .Arithmetic.  New 
A'ork:   Benjamin  H.  Sanborn  and  Company,  1920. 

Thorndike,  Edw.ard  Lee.  The  Thorndike  .Arithmetics.  New  A'ork:  Rand,  Mc- 
Nally  and  Company,  1917. 

Watson,  Bruce  M..  and  White,  Charles  E.  Modern  .Arithmetic  for  Upper 
'Grades.     New  A'ork:  D.  C.  Heath  and  Company,   1918. 

Wentworth,  George,  and  Smith,  D.wid  Eugene.  Essentials  of  .Arithmetic.  New 
York:   Ginn  and  Company,  1915. 

"This  work  was  done  by  Ollie  Asher  during  the  year  1924-25  under  the  immediate 
supervision  of  John  .A.  Clark,  Assistant  in  the  Bureau  of  Educational  Research. 

[48] 


TABLE  II.    NUMBER  OF  PROBLEMS  IN  THE  TEXTS  EXAMINED 


Text 

Number  of  Problems 

Book  II 

Book  III 

Total 

A 

899 
1441 

824 
1339 
1134 
1052 
1269 
1240 
1186 
1070 

876       1775 

B 

1482        2923 

C 

678 

1502 

D 

1953 
1246 

3292 

E 

2380 

F 

1008 

2060 

G 

H 

I 

1336 
1184 
1417 
1363 

2605 
2424 
2603 

J 

2433 

Total 

11454 

12543    i   23997 

It  was  then  classified  under  the  problem  type  described  by  that  ques- 
tion.   (See  page  41.) 

Some  of  the  problems  asked  relatively  simple  questions  but  in  other 
cases  the  question  was  complex  in  the  sense  that  its  answer  involved  the 
specification  of  an  extended  series  of  calculations.  Analysis  of  such 
''complex"  problems  revealed  that  in  most  cases  they  might  be  consid- 
ered as  consisting  of  a  sequence  of  two  or  more  simpler  problems.  Since 
it  soon  became  apparent  that  unless  some  such  policy  were  adopted,  the 
number  of  problem  types  would  be  increased  indefinitely,  the  more 
"complex"  problems,  amounting  to  slightly  more  than  one-fourth  of 
the  total  numbers,  were  classified  as  consisting  of  a  sequence  of  two  or 
more  simpler  problems.  This  procedure  is  illustrated  by  the  following 
problems  whose  classification  is  given  in  the  left-hand  margin.^- 
Bllal(a)  It  cost   Robert  ^4.25   to   grow  the  corn.     He  figures  that   it  cost 

Al  him  9  hours  labor  in  selhng  the  corn.     Counting  the  labor  of  selling 

the  com  at  8  cents  an  hour,  what  was  the  total  cost  of  growing  and 

seUing  the  234  ears? 
A6a2  The   children    of   the    Mullanphy    School    collected   in    two    months 

Blals  11325  lb.  of  old  newspapers  and  2550  lb.  of  magazines.    They  received 

Al  $1.25   per  100  lb.  for  the  newspapers  and  $2.75   per  100  lb.  for  the 

magazines.     What  was  the  total  received  for  old  paper? 
Bllal(a)  Andrew's  father  worked  for  a  farmer.    He  received  $50  a  month  for 

A3  the  12  months  of  the  year,  free  house  rent  worth  $23  a  month,  12  bu. 

Blala  of   potatoes  worth  $1.30  a   bushel,  and   365   qt.   of  milk  worth   8c  a 

Al  quart.     What   he   received  was   equivalent   to  what  money  wages   for 

the  year? 

"For  a  description  of  the  problem  types  indicated  by  the  symbols  used,  see  Ap- 
pendix A. 


[49] 


TABLE  III.  FREQUENCY  OF  OCCURREN'CE  OF  PROBLEM  TYPES 


Book  II 

Book  III 

Total 

Total  frequency  of  problem  types  occurring  in 
simple  problems 

Total  frequency  of  problem  types  occurring  in 
"complex"  problems 

9107 
11801 

8445 
18655 

17552 
30456 

Total 

20908 

27100 

48008 

Blcl  What  percent  of  the  cost  does  a  newsboy  make  on  papers  that  he 

Blf  buys  at  the  rate  of  3  for  4c  and  sells  at  2c  each?     What  percent  of 

B2c2  the  selling  price  does  he  make?    What  percent  of  the  selling  price  does 

AIS^  the  news  dealer  receive?     What  percent  of  the  selhng  price   does  the 

newsboy  receive? 

The  total  number  of  problems  in  each  book  analyzed  is  shown  in 
Table  II.  According  to  Table  III,  17,552  of  the  23,997  problems  were 
classified  under  some  one  of  the  333  problem  types.  The  remaining 
problems  were  considered  "complex"  and  were  classified  as  represent- 
ing a  combination  of  two  or  more  problem  types.  The  fact  that  6,442 
problems  represent  a  total  frequency  of  30,456  problem  types  indicates 
that  most  of  them  were  very  "complex." 

Results  of  the  analysis  of  problem  content.  A  detailed  summary 
of  the  results  of  analyzing  the  ten  series  of  arithmetic  texts  is  given  in 
Appendix  A  in  the  following  form. 

Al  To  find  totals  by  addition,  given  two  or  more  items,  values,  etc. 

A82     B112      C62      D66    El  32      F43      G49      H82      146      J75      749 
A416    B495     C489    D440    E532    F375    G290    H537    1387    J419    4380 

Ble  To  make  change,  given  an  amount  of  money  and  the  price  of  a  commodity. 

A26      B23      C6S      D15      E13  II        Jl       147 

A28      B36      C96      D33      E36  G3        H3      111      J12      258 

The  first  line  of  the  frequencies  gives  the  number  of  occurrences 
of  the  problem  type  when  not  combined  with  another  type,  that  is,  in 
"simple"  problems.  The  second  line  gives  the  total  occurrences  of  the 
problem  type  in  both  "simple"  and  "complex"  problems.  The  number 
of  occurrences  in  "complex"  problems  may  be  found  by  subtracting  the 
upper  number  from  the  lower.  Each  of  the  letters,  A,  B,  C,  D,  E,  F, 
G,  H.  I,  J,  indicates  a  particular  text. 

The  detailed  summary  given  in  Appendix  A  should  be  studied  in 
order  to  secure  a  clear  idea  of  the  nature  of  the  problem  content  of  the 
arithmetics  analyzed.  Only  sixty  out  of  333  problem  types  appear  in 
all  of  the  ten  series  of  arithmetics,  and  only  twenty-five,  ten  or  more 
times  in  all  texts.     A  number  appear  in  only  one  or  two  of  the  texts. 


[50] 


TABLE  IV. 


FREQUENCIES  OF  PROBLEM  TYPES  AND  NUMBER  OF 
PROBLEM  TYPES  IN  EACH  TEXT 


Operation 

Problems 

Activity 

Problems 

Total 

Text 

Total 
frequency 

of 

problem 

types 

Number 

of 

problem 

types 

Total 
frequency 

of 

problem 

types 

Number 

of 

problem 

types 

Total 
frequency 

of 

problem 

types 

Number 

of 

problem 

types 

A 

2602 
3298 
2479 
3046 
2449 
2046 
2782 
3793 
2661 
2613 

44 
35 
39 
38 
38 
35 
40 
37 
44 
37 

1900 
1998 
1564 
2912 
1842 
1805 
2042 
1969 
2059 
2148 

129 
133 
122 
140 
179 
95 
137 
110 
112 
125 

4502 
5296 
4043 
5958 
4291 
3851 
4824 
5762 
4720 
4761 

173 

B 

168 

C 

161 

D 

178 

E 

217 

F     

130 

G 

177 

H 

147 

I 

156 

T 

162 

Total 

27769 

387 

20239 

1282 

48008 

1669 

Twelve  types^^  have  total  frequencies  over  1000.  The  sum  of  the  twelve 
frequencies  is  29,964  or  slightly  more  than  three-fifths  of  the  sum  of  all 
frequencies. 

Table  IV  gives  the  number  of  problem  types  in  each  text  and  the 
sum  of  the  frequencies.  In  interpreting  this  table,  the  reader  should 
bear  in  mind  that  there  are  52  problem  types  under  operation  problems 
and  281  under  activity  problems.  The  fact  that  all  problem  types  do 
not  appear  in  all  texts  is  apparent  from  Appendix  A.  Table  IV  shows 
that  two  texts,  A  and  I.  include  44  of  the  52  problem  types  under  oper- 
ation problems  and  that  35  in  text  F  represents  the  lowest  number  of 
problem  types.  A  somewhat  more  analytical  summary  of  the  operation 
problem  content  of  the  several  texts  is  given  by  Table  IV  A.  It  is  clear 
that  the  texts  differ  in  respect  to  the  problem  types  included  and  also  in 
the  frequency  of  the  occurrence  of  the  types.  For  example,  A14  ap- 
pears in  all  of  the  texts  but  in  text  B  its  frequency  is  1  and  in  text 
H,  50. 

The  variability  among  the  ten  series  of  arithmetics  is  even  greater 
in  the  case  of  the  activity  problems.  Text  F  includes  only  95  of  the 
281  problem  types.  The  greatest  number  of  problem  types  found  in  a 
single  text  is  179  in  text  E.  The  extent  of  the  variability  is  more  clearly 
indicated  by  Table  IVB.    It  is  obvious  in  all  of  the  texts  that  there  are 


"These  with  their  frequencies  are:  Al,  4380;  A2,  4074;  A3,  3532;  A5,  1101;  A6a, 
1556;  A7a,  1148;  AlOa,  1115;  A15,  2172;  A16,  2644;  A19,  1364;  Blal,  4472;  B4a,  2366. 


[51] 


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[S3] 


'•gaps"  in  the  problem  content  of  each  of  the  texts  analyzed,  as  well  as 
variations  in  the  frequency  of  occurrence  of  the  problem  types  included. 
The  vocabulary  of  problems  belonging  to  a  type.  The  reader 
should  not  assume  that  the  problems  classified  under  a  given  type  ap- 
proach identity.  There  are  many  differences  in  vocabulary  and  other 
phases  of  verbal  expression.  The  problems  assigned  to  BlaP"*  were 
examined  and  after  language  duplicates  were  eliminated,  249  different 
problems  remained.  This  examination  revealed  a  large  number  of  ways 
of  stating  the  question  asked.  Some  of  the  more  frequently  used  forms 
are  given  below  :^^ 

Find    the  cost  of  (38) 

the  value  of  (5) 

the  amount  (3) 
How    much  did.  .  .cost  (32) 

much  must . . .  pay  for  (19) 

much  did  .  . .  receive  for  (12) 

much  is  .  .  .  worth  (7) 

much  did  .  . .  get  for  them  (4) 

much  should  be  charged  for  (3) 
What  is  the  cost  of  (30) 

does  . . .  pay  for  (12) 

is  the  value  of  (5) 

is  the   ...  bill  for  (4) 

is  the  exact  cost  (4) 

was  received  for  them  (3) 

was  . . .  worth  (3) 
A  vocabulary  analysis  of  the  249  illustrative  problems  showed  that 
612  different  words  and  phrases  were  used.    Over  one  hundred  of  these 
words  and  phrases  were  used  with  a  technical  meaning.    This  list  in- 
cluded words  such  as: 

all (8)       find (59)       receive (15) 

amount (12)        following (12)        received (7) 

apiece (14)        how  much (98)        selling (9) 

average (11)       paid (12)       sold (33) 

bought (19)       pay (29)       use (5) 

buys (12)        per (41)        used (13) 

cost (149)        price (17)        value (18) 

each (40)        purchases (7)        weighing (7) 

exact (7)        rate (8) 


"■'To  find  the  total  price,  given  the  number  of  units  and  price  per  unit."  The  total 
number  of  "simple"  problems  classified  under  this  type  was  1195. 

*The  numbers  in  parentheses  represent  the  frequency  of  occurrence  in  the  249  il- 
lustrative problems. 

[54] 


Conclusions  concerning  problem  content  of  arithmetic  texts.   The 

data  relative  to  the  problem  content  of  arithmetic  texts  presented  in 
the  preceding  pages  appear  to  justify  the  following  conclusions. 

1.  The  "average"  three-book  series  of  arithmetics  includes  about 
2400  verbal  problems  in  the  second  and  third  books.  This  provides  an 
average  of  600  learning  exercises  per  year. 

In  the  ten  series  analyzed,  the  number  of  verbal  problems  varied 
from  1775  to  3292.  In  six  of  the  series  the  number  of  problems  in  the 
third  book  was  greater  than  in  the  second. 

2.  Although  333  problem  types  were  recognized,  the  average  num- 
ber per  series  was  only  167.  The  lowest  number  included  in  any  ser- 
ies was  130;  the  highest,  217.  Only  60  problem  types  were  found  in  all 
ten  series,  101  were  common  to  eight  or  more  of  the  series,  and  12 
furnished  over  three-fifths  of  the  sum  of  frequencies.  It  is,  therefore, 
obvious  that  the  verbal  problems  of  our  arithmetics  afford  pupils  exten- 
sive opportunity  to  become  acquainted  with  a  few  problem  types, 
(probably  not  more  than  20  to  25)^"  but  beyond  this  limited  number, 
the  opportunities  for  becoming  acquainted  with  functional  relationships 
vary  greatly  with  the  text.  If  the  problems  of  a  series  of  arithmetics 
are  compared  with  the  list  of  problem  types  given  in  Appendix  A,  from 
one  to  two  hundred  of  these  types  will  not  be  found. 


"There  are  only  25  problem  types  which  have  a  frequency  of  10  or  more  for  each 
text. 


tf 


[55] 


CHAPTER  V 

THE    TEACHER'S    RESPONSIBILITY    FOR    DEVISING    AND 
SELECTING  LEARNING  EXERCISES  IN  ARITHMETIC 

General  statement  of  a  teacher's  responsibility.  When  the  imme- 
diate objectives  to  be  attained  by  her  pupils  have  been  estabHshed  and 
a  textbook  adopted,  the  teacher  becomes  responsible  for  devising  or  dis- 
covering in  other  sources  the  learning  exercises  needed  in  addition  to 
those  provided  by  the  text.  She  is  also  responsible  for  selecting  from 
the  available  learning  exercises  those  which  will  provide  the  most  effi- 
cient basis  for  the  attainment  of  the  objectives.  This  second  responsi- 
bility is  especially  important  as  a  means  of  adapting  instruction  to  in- 
dividual differences. 

The  magnitude  of  the  teacher's  responsibility  becomes  apparent 
when  we  recall  that,  although  arithmetic  texts  provide  a  large  number 
of  examples  and  verbal  problems,  there  are  many  "gaps"  in  these  types 
of  learning  exercises  and  that  a  complete  analysis  of  the  textbooks 
would  doubtless  reveal  an  inadequate  supply  of  many  if  not  all  of  the 
types  of  learning  exercises  listed  on  pages  33-35.  It  may  occur  to  the 
reader  that  a  "perfect"  text  would  absolve  the  teacher  of  all  responsi- 
bility in  connection  with  devising  and  selecting  learning  exercises  but 
it  does  not  appear  that  a  millenium  can  be  attained  when  the  same  text 
is  prescribed  for  all  members  of  a  class.  Even  in  a  so-called  homogen- 
eous group,  the  needs  of  all  pupils  for  learning  exercises  will  not  be 
identical.  Their  needs  will  differ  with  respect  to  both  number  and  kind 
of  exercises.  However,  the  more  "perfect"  the  provisions  of  the  text 
and  the  more  homogeneous  the  class,  the  less  the  teacher's  responsibil- 
ity for  devising  and  selecting  learning  exercises. 

Before  attempting  a  more  detailed  statement  of  the  teacher's  re- 
sponsibility for  devising  and  selecting  learning  exercises,  it  will  be  help- 
ful to  note  certain  general  principles. 

k.  Factors  affecting  the  kind  and  number  of  learning  exercises 
needed.  It  is  of  course  obvious  that  in  engendering  a  given  ability  or 
group  of  abilities  the  exercises  assigned  should  be  of  such  a  character 
that  the  resulting  learning  activity  will  produce  the  desired  outcomes. 
Hence  the  immediate  objectives  constitute  the  first  factor  to  be  con- 
sidered in  deciding  the  kind  of  learning  exercises  needed.  If  the  pupil 
is  expected  to  acquire  the  ability  to  do  long  column  addition  without 

[56] 


carrying,  it  is  generally  agreed  he  should  engage  in  the  activity  of  ad- 
ding single  columns  of  figures  of  approximately  the  length  specified  by 
the  objective  and  that  these  examples  should  include  all  of  the  "higher 
decade"  combinations  that  are  included  in  this  objective. 

The  number  of  examples  which  a  given  pupil  should  do  depends 
upon  the  degree  of  facility  to  be  acquired.  If  the  pupil  is  to  acquire 
a  degree  of  skill  equivalent  to  that  possessed  by  an  expert  bookkeeper, 
a  relatively  large  number  of  learning  exercises  will  be  required.  On  the 
other  hand,  if  a  lower  standard  is  set,  for  example,  the  degree  of  skill 
possessed  by  the  average  literate  adult,  a  much  smaller  number  of  ex- 
amples will  suffice. 

The  dependence  of  the  kind  and  number  of  learning  exercises  upon 
the  objectives  is  not  so  apparent  in  all  cases  as  that  noted  in  the  pre- 
ceding paragraph,  but  the  acceptance  of  the  principle  that  learning  is 
an  active  process  implies  the  corrollary  that  a  connection  between  a 
stimulus  or  situation  and  a  response  is  established  only  by  exercising 
it.  Hence,  what  a  child  learns  is  not  the  product  of  learning  activity  in 
general  but  of  the  particular  activities  in  which  he  engages  and  there- 
fore the  learning  exercises  assigned  should  be  devised  and  selected  in 
accord  with  the  recognized  objectives  of  the  subject. 

In  Chapter  I,  the  specific  character  of  arithmetical  objectives  was 
emphasized.  For  example,  it  was  pointed  out  that  the  responses  to 
7  +  5=  ,5  +  7=  ,15  +  7=  ,  and  25 +  7=  involve  the  func- 
tioning of  different  abilities.  However,  such  abilities  are  undoubtedly 
related.  The  pupil  who  is  able  to  respond  fluently  to  5  +  7  ^  will 
more  easily  learn  to  respond  to  15+7^  ,25  +  7=  and  the  like 
than  if  he  did  not  know  5  +  7=  12.  Similarly  the  learning  of  addi- 
tion combinations  facilitates  the  learning  of  the  corresponding  subtrac- 
tion combinations.  Hence,  the  number  of  learning  exercises  necessarv 
to  engender  a  specified  ability  depends  upon  the  amount  of  facilitation 
afforded  by  abilities  previously  acquired. 

The  degree  of  satisfaction  a  pupil  experiences  in  doing  the  exer- 
cises assigned  also  affects  the  number  needed^  and  hence,  indirectly, 
the  kind  that  should  be  assigned.  If  the  exercises  and  the  conditions 
under  which  they  are  done  are  such  that  the  pupil  derives  satisfaction 
from  doing  them,  fewer  exercises  will  be  needed  than  if  the  pupil  is 
indifferent  or  bored  by  the  activity. 

Our  information  concerning  individual  differences  makes  it  clear 
that  some  children  learn  more  rapidly  than  others.  This  means  that 
the  exercise  of  a  connection  between  a  stimulus  and  a  response  produces 

^See  Laws  of  Learning,  pages  31-32,  for  the  basis  of  this  statement. 

[57] 


a  greater  effect  in  some  pupils  than  in  others,  and  hence  the  number 
of  repetitions  required  to  "establish"  a  given  connection  varies  with  the 
capacity  to  learn  arithmetic.  This  is  especially  true  in  the  acquiring  of 
general  and  abstract  meanings.  The  kind  of  learning  exercises  needed 
also  varies  with  the  capacity  to  learn  arithmetic.  Gifted  children  are 
quick  to  grasp  abstract  meanings  and  general  relations  and  hence  do 
not  need  to  do  some  of  the  exercises  that  are  essential  for  pupils  of 
lesser  capacity.  Furthermore,  gifted  children  exhibit  considerable  initia- 
tive and  resourcefulness  in  devising  things  to  do  without  explicit  assign- 
ment. On  the  other  hand,  pupils  of  less  than  average  capacity  to  learn 
and  frequently  those  of  average  capacity  may  require  detailed  explana- 
tions or  demonstrations.  Sometimes  it  will  be  necessary  to  provide 
opportunities  for  perceptual  experiencing. 

A  pupil's  need  for  learning  exercises  at  any  given  period  in  his 
educational  career  depends  upon  his  previous  experience  both  in  and 
out  of  school.  For  example,  it  has  been  found  in  teaching  spelling  that 
a  considerable  percent  of  the  words  specified  by  the  course  of  study 
for  a  given  grade  can  be  spelled  correctly  before  they  are  assigned  for 
study.  Investigations  would  doubtless  reveal  a  somewhat  similar  con- 
dition in  the  field  of  arithmetic,  especially  with  reference  to  general  and 
abstract  meanings.  A  pupil  who  has  lived  on  a  farm  and  become  familiar 
with  terms  such  as  field,  yield  per  acre,  bushel,  pound,  mile,  harvest, 
and  so  forth  has  needs  for  perceptual  experiences  very  different  from 
those  of  a  boy  who  has  never  seen  a  farm.  A  son  of  a  grocer  who  has 
become  familiar  with  his  father's  store,  perhaps  by  working  in  it,  learns 
little  from  dramatizing  a  grocery  store  in  the  school. 
^  Neither  capacity  to  learn  arithmetic  nor  previous  experience  defi- 
nitely known.  Although  we  have  a  number  of  instruments  for  measuring 
general  intelligence,  evidence  indicates  that  capacity  to  learn  arithmetic 
does  not  correlate  perfectly  with  the  capacity  for  learning  in  general.^ 
This  condition,  added  to  the  fact  that  intelligence  test  scores  involve 
errors  (sometimes  relatively  large)  when  considered  as  measures  of 
general  capacity  to  learn,  makes  it  clear  that  these  instruments  will  not 
yield  very  accurate  measures  of  capacity  to  learn  arithmetic.  It  is 
possible  that  more  satisfactory  information  might  be  secured  by  means 
of  a  test  designed  to  measure  capacity  to  learn  arithmetic  but  such 
instruments  have  not  been  constructed. 


^DeVoss,  James  C.  "Specialization  of  the  abilities  of  gifted  children.''  Genetic 
Studies  of  Genii,  Vol.  I.  Stanford  University,  California:  Stanford  University  Press, 
1925,  Chapter  XII. 

[58] 


It  is  likewise  impossible  to  obtain  a  detailed  and  accurate  record 
of  a  pupil's  previous  experience.  Previous  school  records,  particularly 
the  grades  in  arithmetic,  are  indicative  of  school  experience.  Some 
information  concerning  experience  outside  of  school  can  be  obtained 
by  questioning  the  pupil  and  by  ascertaining  the  possibilities  of  his 
environment.  However,  the  information  obtained  will  seldom  if  ever 
approach  a  complete  description  of  a  pupil's  previous  experiences  that 
have  contributed  arithmetical  controls  of  conduct. 

Detailed  determination  of  a  teacher's  responsibility  for  devising 
and  selecting  learning  exercises  a  highly  complex  task.  Recognition 
of  the  factors  affecting  pupil  needs  makes  it  clear  that  the  task  of  deter- 
mining in  detail  the  teacher's  responsibility  for  devising  and  selecting 
learning  exercises  for  a  class  in  arithemtic  is  highly  complex.  Further- 
more, the  varying  provisions  of  arithmetic  texts  make  impossible  a 
detailed  statement  that  would  be  applicable  to  all  situations.  However, 
a  teacher  must  make  a  decision  in  regard  to  the  exercises  to  be  assigned 
to  the  various  members  of  her  class.  To  assist  her  in  this  task  certain 
general  statements  may  be  made. 

I 'The  attainment  of  specific  habit  objectives  requires  repetition. 
When  the  objective  to  be  attained  is  the  formation  of  a  specific  habit, 
the  learning  exercises  must  provide  for  a  considerable  number  of  repeti- 
tions or  practices  of  each  of  the  bonds  connecting  a  response  with  a 
situation  or  stimulus,  such  as  4X8^  32.  The  significance  of  this 
statement  becomes  more  apparent  when  the  range  of  specific  habit  ob- 
jectives is  recalled.  (See  Chapter  I.)  Other  things  being  equal  the 
strength  of  the  connection  increases  with  the  number  of  repetitions  but 
is  also  affected  by  other  factors.  Hence,  in  general  the  number  of  rep- 
etitions necessary  to  attain  the  different  objectives  will  vary. 

Thorndike's  estimate  of  learning  exercises  for  certain  objectives. 
Thorndike^  has  estimated  that  in  the  case  of  an  average  pupil  67  repeti- 
tions of  an  easy  connection*  such  as  "2X5  =  10,  or  10  —  2  =  8,  or 
the  double  bond  7  ^  two  3's  and  1  remainder,"  will  be  sufficient  to 
attain,  by  the  end  of  the  sixth  grade,  the  objective  of  99.5  percent  accu- 
racy in  the  functioning  of  the  connection,  provided  the  repetitions  are 
properly  distributed  and  "the  teaching  is  by  an  intelligent  person  work- 
ing in  accord  with  psychological  principles  as  to  both  ability  and  inter- 
est."   The  distribution  of  the  repetitions   recommended  is   as  follows: 


^Thorndike,  E.  L.  The  Psychology  of  Arithmetic.  New  York:  The  Macmillan 
Company,  1922,  p.  133. 

*The  reader  should  note  that  the  connections  or  bonds  referred  to  are  facilitated 
by  other  bonds  to  a  relatively  high  degree.     This  is  one  reason  why  they  are  "easy." 

[59] 


twelve  during  the  first  week,  twenty-five  during  the  two  months  follow- 
ing and  thirty  spread  over  the  remaining  period.  For  the  more  difficult 
combinations^  100  repetitions  are  estimated  as  necessary.  Gifted 
children  will  require  fewer  repetitions  and  those  who  possess  less  than 
average  capacity  to  learn,  a  larger  number  of  repetitions. 

♦  Isolated  practice  versus  application.  In  considering  the  amount 
of  practice  needed,  it  is  important  to  bear  in  mind  that  repetitions  occur 
in  the  application  of  a  habit  as  well  as  when  it  is  singled  out  for  isolated 
practice.  For  example,  the  addition  combinations  are  exercises  in  doing 
addition  examples  involving  several  figures  to  the  column  as  well  as 
when  the  pupil  responds  to  6  -|-  7  =  ?,  9  -^  8  ^  ?,  13  +  5  ==  r, 
24  +  7  =  .'.  and  the  like.  The  combinations  are  exercised  also  in 
performing  the  calculations  required  in  solving  verbal  problems.  Hence, 
the  total  number  of  repetitions  of  a  particular  combination  is  the  num- 
ber of  repetitions  in  isolation  plus  the  frequency  of  its  occurrence  in 
examples  and  in  the  calculations  required  by  verbal  problems. 

Learning  exercises  needed  for  attainment  of  problem  solving  ob- 
jectives.   In  solving  verbal  problems,  it  is  necessary  that  one  compre- 
hend the  statement  of  the  problem.    This  requires  that  he  must  know 
the  meaning  of  the. technical  words  and  phrases  used,  especially  those  U^ 
which  describe  quantitative  relations.    The  problem  solver  must  detect 
and  comprehend  the  question  or  questions  concerning  functional  rela- 
tionships which  are  implied  in  the  statement  of  the  problem  and  then     ■ 
provide  the  answer  which  will  be  the  specification  of  the  calculations     ^ 
to  be  performed.   It,  therefore,  appears  that  the  problem-solving  objec-    ^ 
tives   include   knowledge   of   the  technical   vocabulary   used   in   stating 
problems   plus   the  ability  to  detect,   formulate  and   answer  questions 
concerning  a  number  of  functional  relationships.^ 

For  responding  to  practical  situations,  a  somewhat  different  com- 
bination of  abilities  is  required.  A  reading  vocabulary  is  not  necessary 
but  one  who  is  efficient  in  dealing  with  practical  situations  probably 
uses  words  as  symbols  of  ideas  and  other  knowledge  elements  in  his 
thinking.    Hence,  mastery  of  a  technical  vocabulary  occupies   a   place 


^There  have  been  several  investigations  to  determine  the  relative  difficulty  of  the 
combinations.     The  most  comprehensive  study  is  by: 

Clapp,  Frank  L.  "The  number  combinations:  their  relative  difficulty  and  the 
frequency  of  their  appearance  in  textbooks."  Bureau  of  Educational  Research  Bulletin 
No.  2.  Madison,  Wisconsin:  University  of  Wisconsin,  July,  1924.  126  p.  In  general, 
the  more  difficult  combinations  are  those  involving  the  large  numbers  but  there  are 
many  exceptions  especially  those  involving  division  combinations  like:  2-^-2=  ,  8-^8=  , 
and  2-=-l=  . 

°See  Appendix  B  for  suggested  minimum  essential  list  of  functional  relationships. 

[60] 


in  the  abilities  required  for  responding  to  practical  situations  Involving 
arithmetical  problems  but  probably  a  less  extensive  vocabulary  suffices 
since  the  problem  solver  is  not  required  to  respond  to  verbal  statements 
formulated  by  another  person.  The  need  for  answering  questions  con- 
cerning functional  relationships  appears  to  be  the  same  for  both  verbal 
problems  and  practical  situations,  provided  the  answer  to  the  problem 
is  to  be  obtained  by  calculation  and  not  by  using  some  mechanical 
device  or  special  rule. 

Although  ability  to  solve  problems  may  include  other  elements, 
knowledge  of  technical  vocabulary  and  ability  to  detect,  formulate,  and 
answer  questions  concerning  functional  relationships  provides  a  basis 
for  certain  observations  concerning  the  learning  exercises  needed  for 
the  attainment  of  the  problem-solving  objectives.  The  outstanding  type 
of  learning  exercise  now  in  use  Is  the  verbal  problem.  Current  practice 
appears  to  assume  that  the  abilities  required  for  solving  problems  can 
be  acquired  efficiently  by  responding  to  verbal  problems,  and  the  pre- 
dominant test  of  the  response  is  the  correctness  of  the  numerical  an- 
swer. Sometimes  the  pupil  is  asked  to  explain,  which  means  that  he 
Is  to  describe  or  give  a  record  of  his  thinking.  A  somewhat  different 
type  of  learning  exercise  is  set  for  the  pupil  when  he  is  asked  to  read 
or  listen  to  an  explanation  of  a  problem.  A  few  other  types  of  learning 
exercises'  are  used  but  usually  only  to  a  very  limited  extent. 

The  nature  of  the  objectives  of  problem  solving  suggests  the  need 
for  exercises  designed  to  provide  an  explicit  basis  for  learning  the  tech- 
nical terms  of  arithmetic.  It  is  true  that  pupils  do  acquire  the  mean- 
ings for  a  number  of  technical  terms  by  solving  verbal  problems  but 
It  appears  that  their  achievements  In  this  field  are  very  inadequate. 
Recognition  of  this  condition  is  evidenced  by  the  common  assertion 
that  the  reason  why  pupils  are  lacking  In  ability  to  solve  problems  Is 
their  inability  to  comprehend  the  statements  of  the  problems.  Vocab- 
ulary tests  have  corroborated  this  belief.^  It,  therefore,  appears  that 
the  reading  of  problems  should  be  made  an  explicit  learning  activity. 
In  testing  this  ability  to  read,  pupils  may  be  asked  to  restate  the  prob- 
lem, substituting  other  technical  terms  when  this  Is  possible;  to  enum- 
erate the  quantities  mentioned  in  the  problem,  specifying  which  ones 
are  given  and  which  one  Is  to  be  found;  to  formulate  the  question  (usu- 
ally implied)  concerning  functional  relationships. 

'These  include  "problems  without  numbers,"  questions  concerning  functional  rela- 
tionships, projects,  collecting  problems  from  newspapers  and  other  sources,  and  the  like. 

'Chase,  Sara  E.  "Waste  in  arithmetic,"  Teachers  College  Record,  18:360-70. 
September,  1917. 

[61] 


The  description  of  the  process  of  solving  an  arithmetical  problem 
as  "detecting  and  formulating  the  question  concerning  a  functional  re- 
lationship, plus  answering  this  question,"  suggests  that  the  more  im- 
portant functional  relationships  should  be  explicitly  taught  as  princi- 
ples. This  will  probably  require  the  use  of  such  questions  as:  "What 
calculations  must  be  performed  to  find  the  area  of  a  rectangle  when 
the  base  and  altitude  (or  two  adjacent  sides)  are  given?"  "How  does 
one  find  the  rate  of  profit,  given  the  cost  of  goods,  and  the  expenses 
and  losses?"  "What  calculation  must  be  made  to  find  the  time  re- 
quired to  travel  a  specified  distance,  given  the  distance  and  rate  of 
travel?" 

"* '  What  problems  should  be  used  as  learning  exercises.  It  is  gener- 
ally agreed  that  problems  differ  in  their  merits  as  learning  exercises. 
Formerly  authors  of  arithmetic  texts  included  many  problems  that  did 
not  occur  in  the  activities  of  adult  life.  Some  of  these  were  verbal  puz- 
zles designed  to  "sharpen  the  wit"  of  the  pupils;  others  were  derived 
from  obsolete  activities;  and  a  third  group,  although  Identified  with  an 
adult  activity,  asked  questions  that  would  never  arise  because  the  an- 
swer would  always  be  known  In  the  practical  situation.  These  kinds  of 
problems  appear  to  be  less  effective  as  learning  exercises  than  real  prob- 
lems, that  is,  problems  simllap^to  those  that  arise  In  the  normal  activi- 
ties of  children  and  adults.  "Similar"  means  not  only  that  the  problem 
Involves  the  same  functional  relations  but  also  that  the  quantities 
(prices,  amounts,  and  so  forth)  be  in  substantial  agreement  with  those 
of  real  life. 

Problems  may  be  real  and  still  differ  in  ways  that  affect  their 
value  as  learning  exercises.  The  analysis  of  ten  texts  described  in  the 
preceding  chapter  showed  the  variety  of  vocabulary  used  and  the  de- 
gree of  complexity  of  many  of  the  problems  appearing  in  current  texts. 
The  desirability  of  an  extensive  technical  vocabulary  Is  largely  a  matter 
of  objectives.  If  the  objectives  of  arithmetic  specify  one  thousand  tech- 
nical terms  to  be  mastered  by  the  end  of  the  eighth  grade,  these  terms 
should  be  used  in  the  stating  of  the  verbal  problems  used  as  learning 
exercises.  If  a  small  number  of  technical  terms  Is  considered  satisfac- 
tory, there  should  be  less  variety  in  the  language  of  the  verbal  problems. 

The  objectives  of  arithmetic  not  yet  determined  in  detail.  The 
objectives  of  arithmetic  were  described  at  some  length  In  Chapter  I. 
Reference  was  made  to  investigations  by  Osburn  and  others  (see  page 
9)  which  indicate  that  the  number  of  combinations  to  be  learned  is 
much  greater  than  commonly  supposed.  Certain  functional  relation- 
ships in  problems  were  listed   (see  page  41)   and  the  vocabulary  used 

[62] 


in  stating  problems  partially  analyzed.  It  was  observed  that  "ability 
to  solve  problems"  means  ability  to  solve  a  problem  that  has  not  been 
solved  before.  Doubtless  the  reader  of  the  preceding  pages  probably 
has  wished  for  a  more  complete  and  definite  exposition  of  the  objec- 
tives of  arithmetic.  With  the  exception  of  the  skills  that  function  in 
calculating,  the  description  of  objectives  was  in  very  general  terms, 
especially  in  the  case  of  those  relating  to  problem  solving.  Even  in  the 
case  of  the  calculation  skills  used  with  integers,  the  objectives  have  not 
been  completely  determined,  and  in  the  case  of  fractions  and  decimals 
less  is  known  concerning  what  a  pupil  should  learn. ^  Hence,  in  at- 
tempting to  determine  the  number  and  kind  of  learning  exercises 
needed  in  the  field  of  arithmetic,  the  assistance  to  be  derived  from 
formulations  of  objectives  is  limited.  However,  the  exposition  of  objec- 
tives in  Chapter  I,  supplemented  by  the  analysis  of  examples  and 
problems  presented  in  Chapter  IV,  should  lead  the  reader  to  formulate 
a  concept  of  the  objectives  of  arithmetic  which  will  be  very  helpful  in 
determining  the  number  and  kind  of  learning  exercises  needed. 

An  estimate  of  the  teacher's  responsibility  for  devising  and  select- 
ing learning  exercises  in  arithmetic.  The  kind  and  number  of  learning 
exercises  needed  in  arithmetic  depend  upon  several  factors  including 
the  objectives,  pupil's  capacity  to  learn  arithmetic,  and  his  previous 
school  experience.  The  preceding  discussion  should  make  it  clear  that 
our  present  state  of  ignorance  about  the  various  factors  makes  it  impos- 
sible to  do  more  than  estimate  the  learning  exercises  needed.  The 
estimate  given  here  is  in  general  terms  and  represents  only  the  writer's 
judgment.  Furthermore,  the  reader  should  bear  in  mind  that  the  details 
of  the  teacher's  responsibility  for  devising  learning  exercises  will  be 
aflfected  by  the  text  adopted  for  use. 

Usually  it  will  be  necessary  for  the  teacher  of  arithmetic  to  devise 
some  exercises  to  provide  perceptual  experiences.  In  the  primary 
grades  opportunities  for  counting,  measuring,  and  estimating  should 
be  provided.  In  the  intermediate  and  upper  grades  the  teacher  may 
require  pupils  to  observe  adult  activities  that  produce  arithmetical 
problems  and  to  give  a  description  of  these  experiences  but  frequently 
it  will  be  advisable  to  ask  the  pupils  to  listen  to,  or  read,  accounts  of 
the  experiences  of  others.   An  important  phase  of  the  devising  of  exer- 


"The  norms  established  for  standardized  tests  constitute  definite  objectives  but 
they  are  available  for  relatively  few  types  of  examples  (see  page  35).  For  a  summary 
of  types  of  examples  for  which  norms  are  available  see: 

Herriott,  \l.  E.  'How  to  make  a  course  of  study  in  arithmetic."  University  of 
Illinois  Bulletin,  Vol.  23,  No.  6,  Bureau  of  Educational  Research  Circular  No.  37. 
Urbana:  Univ^ersity  of  Illinois,   1925.     50  p. 

[63] 


cises  of  the  latter  type  is  the  preparation  or  selection  of  the  descriptions 
that  the  pupils  are  asked  to  listen  to  or  read.  The  authors  of  some 
texts  suggest  construction  exercises,  games,  and  the  dramatization  of 
certain  adult  activities,  but  it  is  usually  necessary  for  the  teacher  to 
plan  at  least  the  details  of  these  types  of  exercises.  In  some  cases  the 
textbook  will  give  little  or  no  assistance.  However,  it  may  be  noted 
that  construction  exercises,  games,  and  the  dramatization  of  adult  activ- 
ities may  not  be  highly  efficient  learning  exercises,  especially  after  the 
primary  grades  are  passed. 

Arithmetic  texts  provide  for  practice  in  reading  and  copying  num- 
bers, supplying  the  missing  number  from  specific  quantitative  rela- 
tionships, and  calculating,  but  analyses  of  the  example  content  of  texts 
show  that  there  are  "gaps"  in  the  practice  on  the  combinations,  both 
basic  and  secondary.  The  teacher  is  responsible  for  discovering  the 
"gaps"  in  the  adopted  text  and  for  devising  the  exercises  necessary  to 
round  out  the  practice.  Usually  the  practice  to  be  provided  is  on  the 
more  difficult  basic  combinations  and  certain  of  the  secondary  combi- 
nations. Several  sets  of  practice  exercises  have  been  devised  to  relieve 
the  teacher  of  a  portion  of  this  responsibility  but  investigation  has 
revealed  that  some  of  the  sets  are  not  satisfactory.  In  some  cases  it 
will  be  necessary  for  the  teacher  to  provide  additional  exercises  In 
reading  and  especially  in  copying  numbers. 

The  "gaps"  in  the  verbal  problems  provided  by  a  text  vary  but 
the  analysis  of  the  problem  content  reported  in  the  preceding  chapter 
Indicates  that  the  teacher  will  need  to  provide  some  additional  prob- 
lems. However,  It  appears  likely  that  the  number  needed  will  not  be 
large  and  pupils  may  be  requested  to  collect  some  of  the  needed  prob- 
lems. On  the  other  hand,  a  number  of  omissions  usually  will  be  justified. 
There  is  no  justification  for  the  very  high  frequency  of  certain  problem 
types  (see  page  51)  and  some  of  the  problem  types  listed  in  the  Appen- 
dix are  not  sufficiently  valuable  to  be  included  In  our  objectives. 

The  teacher  will  need  to  supply  exercises  that  will  lead  to  the 
mastery  of  functional  relationships  (see  page  19),  rules,  definitions, 
and  the  like.  Among  these  are  requests  to  explain,  explanations  to  be 
listened  to,  and  thought  questions.  Other  types  of  learning  exercises 
for  which  the  teacher  must  assume  at  least  some  responsibility  are  re- 
quests to  check  calculations,  to  inspect  and  verify  solutions  of  prob- 
lems, to  collect  quantitative  information,  and  to  generalize  experience. 
Another  very  important  responsibility  of  the  teacher  of  arithmetic  is  to 
provide  exercises  that  will  lead  to  the  mastery  of  the  abstract  and 
general  terms  in  this  field,  especially  those  that  are  used  in  stating  ver- 
bal problems. 

[64] 


APPENDIX  A 

Types  of  Functional  Relations  Found  in  the  Problems  of  Ten 
Series  of  Arithmetics.  The  first  line  of  numbers  gives  the  frequencies 
of  the  simple  occurrences  of  the  relation,  the  second  line  the  frequen- 
cies of  the  total  occurrences  of  the  relation.  For  a  description  of  the 
method  of  analysis  see  page  48. 

A.  OPERATION  PROBLEMS 

Al   To  find  totals  by  addition,  given  two  or  more  items,  values,  etc. 

A82    B112      C62      D66    E132      F43      G49      H82      146      J75       749 
A416    B495    C489    D440    E532    F375    G290    H537    1387    J419     4380 

A2  To  find  the  difl^erence,  given  two  items,  values,  etc. 

A96    B134      C60      D68    E123      F30      G72    H168      131      J50       837 
A395    B648    C370    D445    E429    F186    0345    H532    1314    J410     4074 

A3  To  find  the  amount,  or  number  needed,  by  multiplication,  given  a  magnitude 
and  the  number  of  times  it  is  to  be  taken. 

A53      B96      C20      D36      E75      F47    G107    H161      145      J38       678 
A415    B353    C230    D314    E307    F305    G399    H560    1334    J315     3532 

A4  To  find  the  size  of  a  part  of  a  magnitude,  given  the  magnitude  and  the  number 
of  parts  into  which  it  is  to  be  divided.  (Averages  which  are  the  result  of  division 
only  are  included  here.) 

A28      B36      C39      D34      E30    F44        G86      H53      122      J15       387 
A58      B76    C131      D91      E66    F66      G164    H104      156      J47       859 

A5  To  find  how  many  times  a  stated  quantity  is  contained  in  a  given  magnitude, 
given  the  quantity  and  the  magnitude. 

A16      B59      Cll      D15      E26        F8      G37      H82      129      T26       309 
A91    B148      C72      D57      E94      F70    G187    H154      184    J144      1101 

A6  To  find  how  many  when  reduction  ascending  is  required,  given 

a.  a  magnitude  expressed  in  terms  of  a  single  denomination. 

A6        B9        CI        D2      E20        F6      G27      HIO      116        J7        104 
A163    B125    C118    D220    E115    F151    G191    H125    1167    J181      1556 

b.  a  magnitude  expressed  in  terms  ot  two  or  more  denominations. 

All        B3        CI        D5  F4        G2        H5      113        Jl  45 

A37        B5      C46      D49      E13      F78      G32      H16      145        Jl        322 

A7  To  find  how  many  when  reduction  descending  is  required,  given 

a.  a  magnitude  expressed  in  terms  of  a  single  denomination. 

A6      B20        C2      D17      E16  G14        H5      127      113        110 

A97    B152      C72    D148      E90      F63    G116    HlOl    1145    jl64      1148 

b.  a  magnitude  expressed  in  terms  of  two  or  more  denominations. 

A6        B4        C2        D3        E8        F5        G6        H4      110        J2         50 
A31      Bll      C22      D40      E12      F37      G14      H15      122      J13        217 

A8  To  find  a  dimension,  given 

a.  the  area  of  a  rectangle  and  one  side. 

AlO        B3  D2      E14        F6        G5        H3  J7         50 

AlO        B6        C2        D8      E26      F13      G15        H3        14      J15        102 

b.  the  area  of  a  square. 

Bl  El  G3  12        J2  9 

A2        Bl  Dl        E2        F2        G6  114        J4         32 


[65] 


c.  the  area  of  a  right  triangle  and  one  leg. 

Al 

d.  the  two  legs  of  a  right  triangle. 


A2        B3        C6 

E2        F8 

G7 

HI 

15 

Jl 

35 

A7        B4      C14        Dl 

E4        F9 

GIO 

HI 

112 

J6 

68 

e. 

one  leg  and  the  hypotenuse  of 

a  right  trian 

gle. 

B2        C2 

F3 

G2 

13 

Jl 

13 

A2        B2        C2 

F3 

G4 

14 

Jl 

18 

f. 

the  cubic  contents  of  a  paralle 

lopiped  and 

two  dimensions 

E2        F2 

HI 

17 

J5 

17 

A4        Bl        C5        D5 

E2        F5 

G13 

H4 

110 

J9 

58 

g- 

the  perimeter  of  any  equilateral  figure. 

Bl        CI 

El 

Gl 

11 

Jl 

6 

Al        Bl        C2        Dl 

El 

G2 

11 

P- 

11 

h. 

the  circumference  of  a  circle. 

A2                    CI 

E2        F3 

G6 

11 

Jl 

16 

A2                    CI 

E3        F3 

G8 

14 

J2 

23 

1. 

the  area  of  a  circle. 

G4 

11 

5 

Al 

G9 

HI 

12 

13 

j.    the  cubic  contents  of  cylinder,  and  the  diameter. 

Al  CI        Dl        El  II 

k.  the  cubic  contents  and  area  of  the  base  ot  a  prism  or  cylinder. 
Dl  14 

Dl  H2        15 


Jl 


A9  To  find  a  diameter  equal  to  two  or  more  smaller  diameters,  given  the  smaller 
diameters. 

HI  1 

H2  2 

AlO  To  find  the  area,  given 

a.  dimensions  of  a  square,  rectangle  or  parallelogram. 

All  BIO        C4      D15      E28  F3        G8      H47        19      J14        149 

A73  B118      C72    D166      E85  F106    G161    H159      173    J102      1115 

b.  the  base  and  altitude  of  a  triangle.  (Includes  right  triangle  with   two  legs 
given.) 

E2  G6        H8        II  27 

E5        Fl      G22      Hll        17 


Al        B6        CI        D2 
A7        B7      Cll      D15 

c.   the  diameter  of  a  circle. 


A7        B2        C5        Dl 

d.  the  diameter  of  a  sphere. 
CI 
A3  C2 


El 
E9 


Fl 

F8      G20 


12 

H8      116 


J2 

J6 


Fl 
F4 


G3 


H2 


e.   the  perimeter  of  a  cone  or  pyramid,  and  its  slant  height. 

Gl 


A2  CI        Dl 

f.  the  altitude  and  two  bases  of  a  trapezoid. 

El 
CI        D2        E4        Fl      G12 

g.  the  altitude  of  a  cylinder  and  radius  of  the  base. 

Al 
Al 


HI 


13 
16 

12 
15 

II 

17 


II 


Jl 


5 
20 

2 
10 

2 
29 

1 

2 


[66] 


All   To  find  the  perimeter,  given 

a.  one  side  of  any  equilateral  figure.  •' 

B4    CI    Dl    E5  Gl                  12 

A4    B6    C3    D5    E7  F2    G4         17    J3    41 

b.  two  adjacent  sides  of  a  rectangle  or  parallelogram, 

A5      Bll        C6      DIO        E3  05        H4        18         Tl          53 

All      B21      Cll      D26      E25  G13        H6      124        JT        144 

c.  the  diameter  or  radius  of  a  circle. 

B6        CI        Dl        El  F8        G7                    16                    30 

Al      B15        C6        D4        E9  F18      032        HI      117        15        108 


d.   three  sides  of  a  triangle. 


El  II  2 

El  H3        II  5 


A12  To  find  the  cubic  contents,  given 

a.  one  dimension  of  a  cube. 

Al  D3  4 

b.  1.   the  three  dimensions  of  a   rectangular  solid,  such  as  room,  bin,  wood- 

pile, etc. 
AlO        B3        C8      D15        E6        F5        09      H37      118      Jll        122 
A37      B25      C49      D84      E29      F54      O80      H65      196      J48        567 
2.   the  area  of  one  surface  of  a  rectangular  solid  and  the  depth  or  altitude. 
Al  C2        Dl        E2        Fl  HI        17  15 

c.  the  altitude  and  diameter  or  radius  of  a  cylinder. 

A3        Bl        C3        D4        El  04        H4        14        ]l  25 

AlO        B5      C17      D22        E3      F18      045        H9      118        J4        151 

d.  the  diameter  or  radius  ot  a  sphere. 

H2        II  3 

Al  F4        08        H2        15  20 

e.  the  area  of  the  base  of  a  prism  or  cylinder  and  its  altitude. 

01  1 

Dl        El  Oil  J3  16 

f.  the  area  of  the  base  ot  a  pyramid  or  cone,  and  its  altitude. 

El  1 

Al  CI  El        F4      012  II  20 

A13  To  find  a  difference,  given  denominate  numbers  ot  different  denominations. 

CI  1 

CI  1 

A14   To  find  the  average  given  a  series  of  items.    (To  find  the  average,  given  the  total 
amount  and  the  number  ot  items  is  classified  as  A4.j 

A7        Bl        C7        D3        E6        F3        05      H31        13       113  79 

A8        Bl      C14        D5        E8        F5        05      H50        18      J39        143 

A15   To  find  the  ratio  of  one  number  to  another,  given  the  two  numbers. 

A73      B52      C41    D116    E125      F80      066    H264      153      J54       924 
A199    B183    C225    D327    E193    F175    0162    H418    1151    J139      2172 

.A  16  To  find  a  part  of  a  number,  given  the  ratio  of  the  part  to  the  number  and  the 
number.    (The  fraction  may  be  in  terms  of  fractions  or  decimals.) 

A69      B83      C18    D125      E45      F28      064      H78    1114      T52        676 
A283    B439    C145    D278    E152    FllO    0196    H513    1313    J215      2644 

A17  To  find  a  number,  given  a  part  of  it  and  the  ratio  of  that  part  to  the  whole. 

A24      B69      CIO      D42      E34      F13        09        H5      127       149        282 
A44    B170      C38       D45       E49      F33      026         H7      154    J103        569 


[67] 


A18 


A19 


A20 


All 


A22 


A23 


a:4 


A25 


To  divide  a  quantity  into  parts  having  a  given  ratio,  given  the  quantity  and 
the  ratio. 


A12 
A16 


C5 
C7 


d: 

D3 


Gl 


Hj5 


113 
114 


JIO 
J14 


To  fiud  a  member  of  a  ratio,  given  two  merr^ '->■?--  "• 
of  another  ratio  equal  to  the  first.    (.Inverse  : 

A49    B181      C"6    D116      E80      F:>:     <jl.i  H^> 

A114    B211    Ci:4    D145      E95      F"4    GIjI  Hi5:> 

To  find  the  ratio  of  items  to  total,  given  a  series  of  items. 

Bl  HI 

Al        Bl                                El                    Gl  H7 


atio  and  one  member 


I12I     TllO 

Iloi     T152 


Jl 


982 

1364 


To  compare  pairs  of  quantities  by  ratios,  given  the  pairs  of  quantities. 

C5  E14  G3      H40        15        J6         73 

Al  C20        D4      £15        Fl      Gil      H46        17      J12       117 

To  find  the  largest  quantity  which  will  be  contained  equally  in  two  or  more 

given  quantities. 

B7  7 

B7  7 

To  find  the  least  quantity  which  will  contain  exactly  each  of  two  or  more 
quantities. 

B9 

B9 

To  draw  to  scale,  or  to  represent  graphically  in  tables. 

AS      B14      C39      D42      E23      F31  H^"      131         18       243 

A12      B::      C68      D60      E27      F33  H4^      I3i        J9       311 

To  interpret  tables,  graphs,  or  diagrams,  given  completed  graphs,  tables,  oi^ 
diagrams. 

A26      B19      C61      D23      E30      F26      GI2      H82        17        J7       293 
A30      B21      C"^      D26      E31      F2S      G20      H91        19      J14       34S 


B.  ACTIVITY  PROBLEMS 

Buying  and  selling,^  simple  cases. 
a.  To  find  the  total  price:* 

1.  given  the  number  of  units  and  price*  per  unit. 

A67      B92      C51    D206      E90      F29    GIO"    Hl"9    1212    J162 
A391    B4"3    C348    D625    E433    F289    G4:2    H424    1440    J577 

2.  given  the  number  of  units  and  the  price  per  unit  of  another  denomii 


A20  D3        E8 

A25        B3        CI        D7       E8 
b.  To  find  the  r-umber  of  units: 

1.  given  the  total  price  and  price  per  uait. 
A9      B60      C42      D71      E15      Fll 

A14      B83      C6f      D82      E22      Fi: 


G15 

G2: 


GlO 

015 


H7 

HI  5 


H33 

H55 


19 

117 


133 
138 


JIO 
J27 


J36 
J56 


1195 

4472 
ition. 

72 
130 


320 
442 


I 


■Descriptions  of  quantitative  relations  given  below  are  expressed  in  terms  of 
bu^yiag.  In  some  cases  changes  in  terminology  would  be  necessary  if  the  activity 
were  to  be  considered  from  the  standpoint  of  selling. 

*"Total  price"  is  used  to  designate  the  amount  received  for  several  units  of  the^ 
same  commodity  rather  than  the  amount  received  for  several  commodities. 

'"Price"  is  used  to  designate  the  quanrity  taken  as  a  basis  of  computarioi 
Usually  "price"  refers  to  the  value  or  worth  of  a  unit  rather  than  a  specified  numbei 
of  units.    "Price"  is  often  limited  by  the  qualifying  terms  cost,  selling,  marked,  ant 
list. 


[68] 


2.  given  the  total  price  and  the  price  per  unit  in  another  denomination. 

Jl  1 

Jl  1 

3.  received  in  exchange  of  commodities,  given  an  amount  of  each  commodity 
and  the  unit  for  each. 

C2  2 

C2  2 

4.  given  the  price  per  unit  of  each  of  two  commodities,  the  total  price  of 
both,  and  the  ratio  of  the  number  of  units  of  one  to  the  number  of  units  of 
the  other. 

El  1 

El  1 

5.  given  the  margin,''  per  unit  and  the  total  margin. 

El  ~  1 

Al        B3        C5        D2        E2  Gl  II  15 

c.  To  find  the  price  per  unit: 

1.  given  the  total  price  and  the  number  of  units. 

A19      B22      CIO      D19      E20        F7      G12      H88        19      JIS        221 
A63      B93      C59      D48      E62      F19      042    H163      127      J67        643 

2.  given  the  total  price  and  the  number  of  units  in  another  denomination. 

Al  Gl  2 

A3  Fl        G2  Jl           7 

3.  in  exchange  of  commodities,  given  the  number  of  units  of  each  commod- 
ity  and  the  price  per  unit  ot  one. 

B2  2 

B2  2 

4.  given  the  number  of  units  of  each,  the  combined  price  of  both,  and  the 
ratio  of  the  price  of  the  one  to  that  of  the  other. 

El  1 

El  1 

d.  To  find  the  amount  to  be  received  for  several  items,  given  the  price  of  each. 

A7        B3        C3        D4      E19        F8        G3      H13        II         T3         64 
A22      B22      C15      D37      E72      F43      G28      H28      112      J51        330 

e.  To  make  change,  given  an  amount  of  money  and  the  price  of  a  commodity. 

A26      B23      C68      D15      E13  II         jl        147 

A28      B36      C96      D33      E36  G3        H3      111      J12       258 

f.  To  find  the  margin  or  loss  given  the  cost  price  and  the  selling  price. 

Al        Bl  Fl  H3  Jl  7 

A64    B102      C75      D89      E13      F27      G46      H32      133      J77        558 

g.  To  find  the  total  margin  or  total  loss: 

1.  given  the  number  of  units  and  the  margin  or  loss  per  unit. 

A6  Dl  H3  Tl  11 

A15        B6        C4        Dl      E12        F2        G8        H3        14        J8  63 

2.  given  the  unit  cost,  the  unit  selling  price,  and  number  of  units. 

Al      BIO        CI        D2        E3        Fl        G5  II        J3         27 

A3      B14        CI        D2        E6        Fl        G6        H2        13        J9         47 

h.  To  find  the  margin  or  loss  per  unit,  given  the  total  margin  or  loss  and  the 

number  ot  units. 

Dl  1 

A2        Bl        C3      D30        E2        F3        Gl      H13  55 


^Margin  is  a  term  used  to  represent  the  difference  between  the  cost  price  and  the 
selling  price  and  therefore  is  a  substitute  for  the  words  "gain"  and  "profit"  as  they 
are  commonly  used. 

[69] 


B2   Buying  and  selling,  more  complex  types. 

a.  To  find  the  selling  price: 

1.  given  the  rate*  of  discount  or  loss,  and  the  price. 

A70      B18        C5      D83      E13      F52    G119    H106      141      J14       521 
A82      B25      C14      D86      E31      F64    G128    H140      162      J35        667 

2.  given  the  rate  of  advance  or  margin  and  the  price. 

All      B19        CI      DIO  F6        G8      H25        18        J5         93 

A28      B44      Cll      D13        E8        F8      017      H39      112      J14        194 

3.  given  the  rate  of  two  or  more  successive  discounts  and  the  price. 

A6      B12        C2      D25        E5      F47      037      H15        II      J12        162 
A9      B13        C7      D36      ElO      F49      038      H16      111      J17       206 

4.  given  the  price,  rate  of  advance  or  margin,  and  rate  of  discount  or  loss. 

Al        B2  D2  01  J3  9 

A2        B2  D2        El  G2  J3  12 

5.  given  the  rate  of  commission,  discount,  margin,  or  loss  and  the  amount 

of  commission,  discount,  margin,  or  loss. 

Bl  01  II        J2  5 

Bl  01  II        J3  6 

6.  given  the  price  and  the  amount  of  commission  or  discount. 

E2  2 

A6      B18        C9      D14        E4        F4        09        H5        II      Jll  81 

b.  To  find  the  amount  of  margin,  loss,  commission  or  discount: 

1.  given  the  total  price  and  the  rate  of  margin,  loss,  commission  or  discount. 

A49      B26        C5    D104        E6      F16      029      H34        19      JIO       288 
A91       B76      C30    D140      E28      F30      053      H70      125      J50       593 

2.  given  two  or  more  successive  discounts  and  the  total  price. 

Dl  1 

D4  01  5 

3.  given  the  total  price  and  the  selling  price. 

Al  1 

A3  3 

c.  To  find  the  rate  of  margin,  loss,  discount,  advance  or  commission: 

1.  given  the  total  price  and  the  amount  of  margin,  loss,  discount,  advance 
or  commission. 

A65      Bll        C8        D2      E14      F17        G3        HI        12        J5        128 
A122      B78      C63      D93      E30      F36      O30      H14      123      J60       549 

2.  given  the  cost  price  in  terms  of  two  successive  rates  of  discounts  and  the 
list  price,  and  the  selling  price  in  terms  of  a  single  rate  of  discount  and  the 
list  price. 

116        J2  18 

A2  121        J3  26 

3.  given  the  total  price  and  the  selling  price. 

A24  Cll      D61      ElO      F29      062      H24  T26       247 

A32        B2      C13      D61      Ell      F29      065      H24  J29       266 

d.  To  find  the  price: 

1.  given  the  selling  price  and  the  rate  of  discount  or  loss. 

Al      B17        C2        D2        E5      F13        04  13        J6         53 

A2      B30        C8        D2        E7      F14        04  15      J20         92 

2.  given  the  amount  of  margin,  loss,  commission,  or  discount  and  the  rate  of 
margin,  loss,  commission,  or  discount. 

A8      B19        C7        D2  01        H5  JIO  52 

A12      B34        C8        D2        E2  G3        H5        12      J12  80 


'Rate  may  be  expressed  in  terms  of  percent  or  as  a  fraction. 

[70] 


3.  given  the  selling  price  and  rate  of  margin. 

A2      B24  D2        E4      F29        G6  19        J6  82 

AlO      B37        C5        D2        E5      F30        G7  116      J17        129 

4.  given  the  selling  price  and  two  or  more  successive  discounts. 

^Cl  F2  12  5 

Bl        CI  El        F2  12  7 

e.  To  find  the  amount  due  the  agent  or  agents,  given  the  number  of  units,  the 
price  per  unit,  and  the  rate  of  commission.    (Agent  purchases  commodity.) 

Al      BIO        C3  G2  12  18 

Al       B12        C4  02  12  21 

f.  To  find  the  equivalent  single  discount  in  percent,  given  two  or  more  successive 
rates  of  discount. 

CI  El  2 

CI  El  2 

g.  To  find  one  of  two  or  more  successive  discounts,  given  the  list  price,  one  or 
more  of  the  successive  discounts  in  percent,  and  the  net  price. 

Jl  1 

Jl  1 

B3   Carrying  on  a  business. 

Note:  Types  listed  under  "carrying  on  a  business"  are  similar  in  certain  respects 
to  those  found  under  the  activity  ol  "buying  and  selling,"  but  in  general  the  follow- 
ing distinction  prevails.  The  problems  under  Bl  and  B2  are  those  in  which  a  pur- 
chaser is  explicitly  involved  and  in  which  he  may  be  expected  to  be  interested,  at 
least  to  the  extent  of  checking  the  solution  by  the  seller.  The  problems  under  B3 
are  those  which  in  general  only  the  one  carrying  on  the  business  will  encounter.  The 
degree  of  magnitude  of  the  quantities  of  the  problem  and  the  terminology  were  also 
used  as  criteria.  In  cases  where  the  distinction  is  not  obvious,  a  footnote  indicates 
similarities  or  differences. 

a.  To  find  the  selling  price:^ 

1.  (a)   given  the  cost  price,  rate  of  net  profit,  and  rate  of  overhead.    (Profit 

and  overhead  are  figured  on  the  cost  price.) 

HI  1 

HI  1 

(b)given  the  cost  price,  rate  of  net  profit,  and  rate  of  overhead.     (Profit 
and  overhead  are  figured  on  selling  price.) 

El  1 

El  I 

2.  (a)   given  the  cost  price,  rate  of  net  profit,  and  the  overhead.    (Profit  is 

figured  on  cost.) 
A4  Dl  Gl  6 

A6        B4  D4  G2  Jl  17 

(b)   given  the  cost  price,  rate  ot  net  profit,  and  the  overhead.    (Profit  is 
figured  on  selling  price.) 

Bl        CI  El  3 

b.  To  find  the  total  receipts,  given  the  total  cost  and  the  net  profit. 

El  I 

c.  To  find  the  overhead,  given  the  cost  price  or  selling  price  and  the  percent  of 
overhead. 

E6  G4  10 

CI  E17  G9  II  28 

d.  To  find  the  rate  of  overhead,  given  the  cost  price  or  selling  price  and  the 
overhead. 

G4  Jl  5 

^General  terminology  and  "overhead"  are  the  factors  which  distinguish  this 
classification  from  B2a. 

[71] 


e.  To  find  the  net  profit  or  loss: 

1.  given  the  cost  price,  overhead,  and  selling  price- 

El  H3  4 

A5        B7        CI        Dl      E20        F2        G6      HIO        16  58 

2.  given  the  cost  price,  rate  of  overhead,  and  selling  price.     (Overhead  is 
figured  on  the  cost.) 

DIO  10 

Bl        C2      DIO        E3  Gl  17 

3.  given  the  cost  price,  the  rate  of  overhead,  and  the  selling  price.    (Overhead 
is  figured  on  the  selling  price.) 

E4  4 

A2        Bl        CI  E4        F3        04  15 

4.  given  the  total  costs  and  •■otal  receipts. 

El  Gl        HI  3 

A8        B8        C9      D17      E25        Fl      G21        H8        14      JIO        111 

5.  given  the  itemized  costs  and  total  receipts. 

Al    B3  E3  HI  8 

A4   B9   C6   D12   E12   F2        H2   12         49 

6.  given  the  rate  of  gross  profit,  the  rate  of  overhead,  and  the  selling  price. 
(Overhead  and  gain  are  figured  on  the  selling  price.) 

G5  5 

El  G5  6 

7.  given  the  rate  ot  gross  profit,  the  overhead,  and  the  selling  price.    (Gross 
profit  is  figured  on  the  selling  price.) 

G2  2 

G3  3 

8.  given  the  gross  profit  and  the  overhead. 

G2  Jl  3 

f.  To  find  what  percent  the  net  profit  is  ot  the  cost  price  or  selling  price,  given 
the  cost  price,  overhead,  and  the  selling  price. 

Al  E2  G3  6 

A5  E4  G4  Jl         14 

g.  To  find  what  percent  the  net  profit  is  of  the  cost  price  or  selling  price,  given 
the  net  profit,  and  the  total  receipts,  original  outlay,  or  amount  invested. 

Ell      F38        G2  51 

B4        CI        D8      E31      F41        G6  Jl  92 

h.  To  find  what  percent  the  profit  or  loss  is  of  the  cost  price  or  selling  price, 
given  the  profit  or  loss  and  the  cost  price  or  selling  price.' 

Bl  H2        II  4 

A3        B7        C4        Dl        E9        Fl        G7        H6        II  39 

i.    To  find  the  gross  profit: 

1.  given  the  cost  price  and  total  receipts. 

ElO  Gl  11 

2.  given  the  cost  price  or  selling  price  and  rate  of  gross  profit. 

G4  Jl  5 

j.    To  find  the  rate  of  gross  profit: 

1.  given  the  wholesale  price  (cost  price)  and  the  retail  price  (selling  price). 

El  Jl  2 

E2  .  Jl  3 

2.  given  the  total  receipts,  total  amount  invested  or  total  costs,  and  the 
amount  of  gross  profit  or  gross  income. 

Bl  E6  G2  9 


^This  classification  differs  from  B2cl  in  size  of  quantitative  terms  and  in  termi- 
nology. 

[72] 


k.  To  find  the  amount  invested: 

1.  given  the  itemized  costs. 

El  1 

Dl         E3  Jl            5 

2.  given  the  rate  of  profit  on  the  investment  and  the  net  profit. 

Bl                                            Fl                                 II  3 

Bl                                El        F2        Gl                    12  7 

I.    To  find  the  profit  on  an  investment,  given  the  amount  of  the  investment,  the 
rate  of  profit  per  unit  of  time  and  the  time. 

Bl                                              Fl                                138  40 

B2                                El        F2        Gl                   141  47 

m.  To  find  the  profit  per  person  on  the  basis  of  investment,   given  the   amount 
invested  by  each  person  and  the  profit  on  the  total  investment. 

Bl        CI  2 

B2        CI  3 

n.  To  find  the  commission: 

1.  given  a  series  of  commodities,  the  number  of  units  in  each  series,  the  unit 
price  of  each  commodity,  and  the  rate  of  commission. 

CI                    El                   Gl  3 

A5                   CI                    El                   Gl        HI  9 

2.  given  the  total  sales,  the  expenses,  and  the  rate  of  commission  charged. 

Bl  1 

Bl                                                                    HI  2 

3.  given  the  total  cost,  the  expenses,  and  the  amount  remitted  to  the   agent 

Dl  1 

Dl  1 

4.  given   the  proceeds  remitted   by  the  agent,  the  rate  of  commission,  and 
expenses. 

Al  1 

Al  1 
o.  To  find  the  vnlue  of  goods  to  be  sold,  given  the  rate  of  commission  and  the 
amount  of  commission  to  be  earned. 

Al         B3                                  El  J2            7 

Al        B4                                E2  J5          12 

p.  To  find  the  amount  remitted  to  the  agent  given  the  selling  price,  the  rate  of 
commission  and  expenses. 

Gl  1 

Al                    CI         Dl        El                    G3  Jl            8 
q.  To  find  the  value  of  goods  sold  (selling  price),  given  the  net  proceeds,  the 
rate  ot  commission  and  the  expenses. 

B2                                              Fl  J3            6 

B2                                              Fl  J3            6 
r.   To  find  the  net  proceeds.    (Wholesaler's  point  of  view.) 

1.  given  the  amount  of  the  sales  and  the  rate  of  commission. 

B2        CI      D30        El        F7                                II  42 

B3        C3      D32        E3      FIO                                12  53 

2.  given  the  amount  of  sales,  rate  of  commission  and  expenses. 

Al        B4                                E2                   Gl  J2          10 

A2      B16        C7        D3        E4                    G6        HI  J3         42 

3.  given  the  commission,  rate  of  commission  and  expenses. 

Al        Bl  2 

Al        Bl  2 
s.  To  find  the  rate  of  commission: 

1.  given  the  cost  price  or  selling  price,  the  expenses,  and  the  amount  remitted 
to  the  agent. 

Al  Dl  2 

[73] 


2.  given  the  proceeds  and  the  amount  of  sales. 

El  I 

E2  Fl                                                          3 

3.  given  the  amount  remitted  to  the  owner  or  dealer,  the   amount  of  sales, 
and  the  expenses. 

Al  1 

Al                                  Dl  2 

B4   Borrowing,  lending,  and  saving  money. 

a.  To  find  the  interest  or  discount,  given  the  amount  loaned,  the  rate  of  Interest 
or  discount,  and  the  time  or  term. 

A183      B68      C57    D343      E70    F394    G234    H168    1346    J251      2114 
A195      B98      C79    D361      E86    F426    G245    H232    1374    J270     2366 

b.  To  find  the  total  interest  received,  given  the  rate  of  Interest,  the  amount 
loaned,  the  term  for  compounding  the  interest,  and  the  total  time. 

B2   CI    Dl    E9  J4    17 

A13    B2    C4    Dl    E9  J4    33 

c.  To  find  the  exact  interest,  given  the  date  the  loan  was  made,  the  date  the 
loan  was  due,  the  loan,  and  the  rate  of  interest. 

All        B5        C2        D2        E8      F31  J6  65 

All        B5        C3        D2        E8      F31  J6  66 

d.  To  find  the  amount  loaned: 

1.  given  the  interest  or  discount,  the  rate,  and  the  time. 

Al        B2        C2        D5      ElO      FIO                    H5        15  J3  43 

Al        B4        C2        D5      ElO      F12        G3        H7        15  J3  52 

2.  given  the  amount  due,  the  time,  and  the  rate  of  interest. 

B2                                 E5        F2  Jl  10 

B2                                E5        F2  Jl  10 

e.  To  find  the  face  of  a  note,  given  the  rate  of  discount,  the  proceeds,  and  the 
term  of  discount. 

E2  J5  7 

E2  J6  8 

f.  To  find  the  amount  due: 

1.  given  the  amount  loaned,  the  rate  of  Interest,  and  the  time. 

A2      B48      C17    D120      E66        F6      G18        H6      110  J13  306 

A9      B70      C43    D122      E71        F9      G26      H19      139  J29  437 

2.  given  the  amount  loaned,  the  rate  of  discount,  and  the  time. 

E7        F8  J2  17 

CI                    E7        F9                    H6  J2  25 

3.  given  the  rate  of  Interest,  the  amount  loaned,  the  term  for  compounding 
the  interest,  and  the  total  time. 

A5  D3      E19      F22        G5      HIO      121        Jl  86 

A5        B2        C3         D3      E20      F25        G6      HIO      121        Jl  96 

4.  at  a  given  time,  given  one  or  more  deposits,  the  date  of  each  deposit,  the 
rate  of  interest,  and  the  term  for  compounding  the  interest. 

AlO        Bl      Cll      D20        E3        Fl      G15  12  63 

AlO        B3      C12      D21        E3        F2      G15  13  69 

g.  To  find  the  balance  due,  given  the  amount  loaned,  the  time  of  interest  pay- 
ments, the  partial  payments,  the  total  time,  and  the  rate  of  interest. 

A6  C2        D5  Gl         HI         16       113  34 

A6        B3        C2        D5  Gl         HI         16      }l4  38 

h.  To  find  the  proceeds,  given  the  face  of  the  note,  draft,  or  trade  acceptance, 
term  of  discount,  rate  of  discount  and  time. 

All        B4        CI      D21      E20      F15      G28  163       115        178 

A17      B20      C15       D34      E22      F17      G29  172      J20        246 

[74] 


i.  To  find  the  balance  due  at  a  given  time,  given  a  series  of  deposits,  the  time  of 
each  deposit,  withdrawals,  rate  of  interest,  and  the  term  for  compounding 
the  interest. 

A3  C3        Dl  7 

A3  C3        Dl  7 

j.  To  find  the  rate  of  interest  or  discount,  given  the  amount  loaned,  the  amount 
of  interest  or  discount,  and  the  time. 

CI         D9      E14      F61  H8        16         ]2        101 

CI      DIO      E15      F63  H14        18        }3        114 

k.  To  find  the  time,  given  the  amount  loaned,  the  rate  ot  interest,  and  the 
amount  of  interest.    (Reductions  of  time  elements.) 

Al  D9        E9        Fl  II  21 

Al        B3  D9      Ell        Fl  II  26 

B5    Keeping  accounts. 

a.  To  find  the  total  of  a  bill  or  invoice,  given  an  Item  or  series  of  items,  the 
number  of  each,  the  price  of  each,  and  the  terms. 

A4      B13        C7      D15      E41        F7        G3        H9      124        J2        125 
A17      B16      C15      D30      E46      F31        G3        H9      125        J5        197 

b.  To  find  the  total  value,  given  an  inventory,  and  value  of  each  item. 

^  Dl         El  H2  4 

D2        El  H2  5 

c.  To  make  out  a  bill  or  invoice,  given  an  item  or  a  series  of  items,  the  number 
of  each,  the  price  of  each,  the  names  of  the  purchaser  and  seller,  and  the  terms. 

A2        B7        C4        D5        El      F14      G20        HI      145        J2        101 
A14      B35      C15      D51        E8      F38      G38        H2      163      J14       278 

d.  To  make  out  a  bank  deposit  slip,  given  two  or  more  checks,  an  amount  of 
bills,  and  several  coins. 

A4  C2  E4        F6        G3  18  27 

A4  C2        Dl        E4        F6        G3  111  31 

e.  To  make  a  monthly  statement,  given  the  items  bought,  the  credits  allowed, 
the  purchaser,  and  the  seller. 

C3  II  4 

C3        D3  II  7 

f.  To  make  a  contract,  given  the  agreement,  the  consideration,  the  parties  con- 
cerned, and  the  witnesses. 

C2  El  3 

C2  El  3 

g.  To  make  out  an  inventory,  given  a  series  of  items,  the  number  in  each  series, 
and  the  value. 

HI         II  2 

HI         II  2 

h.  To  receipt  a  bill  when  paid,  given  the  bill  and  the  payment. 

Jl  1 

A8      B27        CI      D39        E7        Fl      G14        HI      112      J13        123 

i.  To  write  a  receipt,  given  the  amount  for  which  payment  was  received  and 
the  name  of  the  payer. 

A4        B3      C12        D6        E5  HI        17  38 

A4        B5      C15        D6        E5  HI        18  44 

j.  To  write  a  note  or  trade  acceptance,  given  the  amount,  rate  ot  interest,  payee, 
payer,  and  time. 

A7        B4        C3        D2        E2        F8        Gl        H4        II  32 

AlO        B6        C8        D4        E2      F14        G3        H4      115        J5  71 

k.  To  write  a  check,  given  the  name  of  the  bank,  the  amount  of  the  check  and 
the  payee. 

A5        Bl        C5        D4  G2  17 

A7        Bl      C17      D17        El  G4        H4        II  52 

[75] 


1.  To  write  a  draft,  given  the  amount  of  the  draft,  the  name  of  the  person  in 
favor  of  whom  the  draft  is  drawn,  of  the  bank  on  which  the  draft  is  drawn, 
and  the  bank  drawing  the  draft. 

A2  C3        D4        El  G3  13        J3  19 

A2  C3        D4        E2  G4  17        J4  26 

m.  To  keep  a  stub  of  a  check  book: 

1.  given  an  original  deposit  and  a  series  of  checks. 

GI  1 

Gl        HI  2 

2.  given  an  original  deposit,  a  series  of  checks,  another  deposit,  and  another 
series  of  checks. 

E3  G6  9 

E3  G6  9 

n.  To  keep  a  cash  book,  given  receipts  and  expenditures. 

Bl        C2  Gl        HI  14 

A5        B2        C8        D4        E2        F6        G8        H2      122        J4 

o.  To  keep  an  account,  given  purchases  and  payments,  simple  accounts. 

CI  F6        G3 

p.     To  indorse  a  check,  given  a  check.    (Drafts  and  notes  included.) 
A2  CI        D2  F2        Gl 

A6  CI        D6        El        F8        G4  19        J4 

q.  To  find  the  balance  of  a  cash  book,  given  expenditures  and  receipts. 
A2        Bl        C6      D15      E22  G7        H2 

AS        B2      C12      D19      E26      Fll      G15        H3      122        J2 
r.   To  balance  an  account,  given  purchases  and  payments. 

1.  simple  accounts. 

E2        Fl        G6 
CI        D3        E3        F7        G9 

2.  complex  accounts. 

G5 
G5 

s.  To  balance  a  bank  account,  given  an  original  balance,  a  series  of  deposits,  and 
a  series  of  withdrawals. 

A2  CI        Dl  F5        G7        H7      110        Jl  34 

A2  CI        D3  F5        G7        H7      114        Jl  40 

t.   To  find  a  balance,  given  the  exchange  of  commodities. 

Jl  1 

Jl  1 

B6   Construction. 

Note:  Problems  involved  in  the  following  activities  were  included  in  this  classi- 
fication: woodworking,  sewing,  cooking,  building  construction,  and  fencing.  Costs  of 
construction  materials  were  included. 

a.  To  find  how  many  times  a  given  pattern,  border,  design,  or  length  is  con- 
tained in  a  given  length. 

A7      B15        C7        D7      E16  G38      H13        19        J3        115 

A26      B27      C15      D45      E18        F4      G51      H20      118        ]3       111  \ 

b.  To  find  the  amount  of  fencing,  given  the  number  of  wires  to  be  used  in  a 
dimension  of  the  area. 

Gl 
Bl  Gl 

c.  To  find  the  total  number  ot  units: 
1.  given  the  dimensions  of  the  unit,  and  the  dimensions  of  the  whole 

A24      B17      C12      D21      E22        F9      G48        H8        18      J27        196] 
A37      B28      C20      D44      E37      F30      G84      H13      117      J59       3691 


9 
63 


10 


39 

55 
120 


9 

23 

5 

5 


[76] 


2.  given  the  number  of  wholes  and  the  dimensions  of  each  whole. 

Gl        HI  2 

B2  G8        HI  J5  16 

3.  given  the  dimensions  of  the  whole  and  the  size  of  the  unit. 

Al  G9  Jl  11 

Al        B3  Dl        E6  G14  II        J6         32 

4.  given  the  dimensions  of  the  whole,  the  size  of  the  unit,  and  the  allowance. 
(Allowance  for  openings,  waste,  matching,  etc.) 

13  3 

B3  Dl        El  G3  110        J6         24 

5.  given  the  dimensions  of  the  whole,  the  dimensions  of  the  unit,  and  allow- 
ance for  waste,  etc. 

F3  II  4 

CI        D3        El        F3  15        Jl  14 

d.  To  find  the  number  of  shingles  needed: 

1.  given  the  number  of  shingles  used  per  square  or  a  given  area,  and  the 
dimensions. 

El        F2  12  5 

Dl        El        F2  12        Jl  7 

2.  given  the  number  of  shingles  used  per  square  and  the  area  to  be  covered. 

El  1 

El  Jl  2 

e.  To  find  the  total  number  of  board  feet,  given  a  mill  bill. 

A2  D42  H25        18  77 

A5        B3        CI      D49        E4        F7  H36      113      J23        141 

f.  To  find  the  amount  of  paint  needed  to  cover  an  area,  given  the  area  covered 
by  a  unit  measure  of  paint  and  the  total  area  to  be  covered. 

Bl  Gl  2 

B2  E4  Gl  7 

g.  To  find  the  number  of  rolls  of  paper  needed,  given  the  dimensions  of  the 
room,  and  the  allowance  for  openings. 

F5  5 

D6  F5  11 

h.  To  find  the  rim  speed,  given  the  number  of  revolutions  per  minute,  and  the 
diameter. 

H14         12  16 

H14        12  16 

i.    To  find  the  number  of  revolutions  per  minute,  given  the  rim  speed,  and  the 
diameter. 

Gl        H3  4 

A3  Gl        H3        II        Jl  9 

j.    To  find  the  total  cost  of  construction,  given  the  cost  per  unit  and  the  number 
of  units. 

R 1  H 1  16  8 

A23      B47        C4      D50      E27      F19      G33        H4      124      J34       265 
k.  To  find  the  cost  per  unit  of  construction,  given  the  total  cost  and  the  number 
of  units. 

Al        Bl  El  Gl  13  7 

A3        Bl  Dl       El        F3        Gl        HI        13  14 

1.    To  find  the  number  of  units,  given  the  total  cost  and  the  cost  per  unit. 

E4  4 

E4  4 

m    To  find  the  number  of  units,  given  the  size  of  the  whole,  and  the  size  of  the 
unit. 

Bl  HI  2 

Bl  E9  HI  II 

[77] 


B7  Travel,  transportation,  and  communication. 

Note:  This  type  of  problem  includes  travel  by  any  means  such  as  automobile, 
train,  etc.  It  also  includes  transportation  by  truck,  train,  express,  parcel  post,  or  by 
any  other  means.  Communication  of  any  type  may  be  included  here,  such  as  mail, 
telephone,   telegraph,   or   radio. 

a.  Travel. 

1.  To  find  the  distance: 

(a)  given  the  time  and  the  rate. 

Bll        C2        D8      E13        F5        G4      H19        16        Jl  69 

A2      B23        C4        D8      E17        F9        G4      H29        19        J6        111 

(b)  between  two  places,  given  the  rate  of  travel,  the  time  taken  to  travel 
the  distance,  the  number  of  stops  and  the  time  for  each  stop. 

Dl  1 

Dl  1 

2.  To  find  the  distance  traveled  per  unit  of  time: 

(a)  given  the  total  distance  and  the  total  time. 

A2        B3        C5      D13        E7      F14      G16      H20        12        J2 
A3        B7      Cll      D17      ElO      F30      022      H25        18        J3 

(b)  given  the  distance  between  two  places,  the  time  taken  to  travel  the 
distance,  and  the  time  spent  for  stops. 

01 


84 
136 


01 


HI 
HI 


n 


3.   To  find  the  time: 

(a)  given  the  distance  and  the  rate. 

B4        C4        D2      ElO        F5        05        H5        18        Jl  44 

B7        C4        D2      E17        F7        05        H8        18        J2         60 

(b)  between  two  stations,  given  one  station  in  one  time  belt  and  another 
station  in  another  time  belt.    (Eastward  travel.) 

Bl  1 

Bl  1 

(c)  between  two  stations,  given  one  station  in  one  time  belt  and  another 
station  in  another  time  belt.     (Westward  travel.) 

Bl  1 

Bl  1 

b.  Transportation. 

1.  To  find  the  amount  hauled  by  the  same  power  over  a  good  road,  given  the 
power,  the  amount  hauled  on  a  poorer  road,  and  the  ratio  of  the  amount 
hauled  on  the  poorer  road  to  that  which  can  be  hauled  on  a  better  road. 

El  1 

El  1 

2.  To  find  the  number  of  trips  needed  to  haul  a  given  amount  over  a  good 
road  with  the  same  power  used  on  a  poorer  road,  given  the  power,  the 
amount  to  be  hauled,  the  amount  hauled  per  load  on  the  poorer  road  and 
the  ratio  of  that  load  to  the  load  hauled  on  the  better  road. 

E2  2 

Dl        E3  4 

3.  To  find  the  cost: 

(a)  of  sending  a  commodity  or  commodities  by  parcel  post,  given  the  rate 
of  the  article  for  a  given  zone,  and  the  weight. 

A26        B7      C27      D22        El       F22        01  ]3        109 

A27        B7      C28      D22        El      F22        Ol  13      J16        127 

(b)  of  shipping  commodities  by  express,  given  the  rate,  the  weight,  and 
distance. 

A6        B2        C2        D2      Ell  H  24 

A9        B4        C7        D5      El  2        Fl  HI         II        J2  42 


[78] 


(c)  of  shipping  small  commodities,  given  the  cost  per  pound,  weight,  or 
size. 

A4  4 

A4  4 

(d)  of  hauling  bulk  commodities,  given  the  total  number  of  units  and  the 
cost  per  unit. 

11  1 
Bl        CI        Dl        El                                            15                       9 

(e)  of  carrying  a  load  of  equal  weight  over  a  good  road,  given  the  cost  of 
power  per  mile  on  a  poorer  road,  the  distance  traveled,  and  the  ratio 
of  the  load  the  same  power  can  haul  on  the  good  road. 

El  1 

El  1 

(f)  To  find  the  cost  per  unit  of  hauling  or  transportation,  given  the  total 
cost  and  the  number  of  units. 

Dl  12  3 

A3  C3        Dl  15  12 

4.  To  find  the  total  cost  of  an  article  sent  by  parcel  post,  given  the  weight, 
the  rate  for  the  zone,  the  value  of  the  article  or  articles,  and  the  rate  of 
insurance. 

C2  El  3 

C15  El  16 

5.  To  find  the  freight  rate,  given  the  amount  of  freight  charges,  and  the 
weight. 

CI  1 

CI        Dl  2 

c.    Communication. 

1.  (a)   To  find  the  amount  charged  lor  collection  of  a  draft,  given  the  face 

value,  and  the  rate  charged. 

D3        E2  J5  10 

D3        E5  J7  15 

(b)  To  find  the  rate  of  exchange,  premium,  or  discount,  given  the  face 
value  of  a  money  order  or  draft,  and  the  total  cost. 

12  2 
12  2 

(c)  To  find  the  proceeds,  given  the  amount  ol  the  draft,  money  order,  or 
bill,  and  the  rate  charged  for  collection. 

E2  II  3 

E8  119 

2.  To  find  the  cost: 

(a)  of  a  money  order  or  draft,  given  the  amount  sent,  and  the  rate  charged. 

CI      DIO      E12        F6  H2      145      116         92 

A17  CI      DIO      E19        F6        Gl        H2      145      J26        127 

(b)  of  mailing  letters,  newspapers,  etc.,  given  the  rate  of  postage  per  unit 
and  the  number  ot  units.    (Unit  may  mean  letters  or  weight.) 

A14  D2  F2  18 

A20  Dll  F4  35 

(c)  To  find  the  cost  of  sending  a  telegram  or  telephoning,  given  the  number 
of  units,  a  rate  lor  a  given  number  ot  units,  and  an  added  rate  for  each 
additional  unit. 

Bl  D3  4 

Bl  D3  4 

B8   Municipal  and  federal  activities.    (Excluding  taxation.) 
a.   Municipal  activities. 

1.  To  find  the  per  capita  expense  of  a  community  activity,  given  the  total 
cost  and  the  population.    (Total  number  of  persons.) 

E4      F12        Gl        H2        II  20 

CI  E6      F12        Gl        H2        II  23 

[79] 


2.  To  find  the  number  of  lives  saved,  given  the  death  rate,  the  decrease  in 
percent  (due  to  an  applied  remedy)  and  the  population. 

El  1 

El  1 

3.  To  find  the  number  of  lives  saved,  given  the  death  rate  at  one  period,  the 
death  rate  at  a  later  period,  and  the  population  at  each  period. 

El  1 

4.  To  find  the  per  unit  cost  of  a  community  activity,  given  the  total  cost, 
and  the  number  of  units. 

A2  G7  II  10 

A2  Dl  F4        G7  II  15 

5.  To  find  the  per  capita  loss,  given  the  valuation  of  property  destroyed,  and 
the  total  population. 

El  1 

6.  To  find  the  total  cost  of  a  community  activity,  given  the  number  of 
units  or  the  total  population,  and  the  cost  per  unit  or  per  person. 

EI  G2  3 

E2  02  4 

7.  To  find  the  death  rate  per  a  given  number,  given  the  number  of  deaths, 
and  the  total  number  of  persons. 

CI  E3        Fl  5 

C17  E3        Fl  21 

b.  Federal  activities. 

1.  To  find  the  number  of  years  of  peace  needed  to  pay  for  a  year  of  war, 
given  the  amount  saved  during  a  year  of  peace,  and  the  total  amount 
spent  during  a  rear  of  war. 

HI  1 

B9   Insurance. 

a.  To  find  the  premium: 

1.  given  the  face  value  of  the  policy,  the  rate  of  insurance,  and  the  term. 

A4        B5  D22  F4        G2        H9      136        Jl  83 

A19      BIO        CI       D29        El       F13        G2      H12      137        J9        133 

2.  given  the  valuation  of  the  property,  the  ratio  of  that  value  which  was 
accepted  for  insurance,  rate  and  term. 

Al        Bl        C3        Dl        E5        Fl        Gl        H2        14        J3         22 
AS        Bl        C5        D2        E6        Fl        G5        H2        15        J5         37 

3.  given  the  face  value  of  the  policy,  the  original  rate  of  insurance,  the  per- 
cent of  decrease  due  to  the  installation  of  protection  devices. 

El  ■  1 

b.  To  find  the  total  premium,  given  itemized  values  and  respective  rates,  and  an 
added  rate  for  an  additional  risk. 

El  1 

El  1 

c.  To  find  the  amount  of  insurance,  given  the  rate  and  the  premium. 

B3        CI  Gl  114        J8  27 

B3        C2  El  Gl  116      J12         35 

d.  To  find  the  rate  of  insurance,  given  the  premium,  the  face  value  of  the  policy, 
and  the  term. 

Al  D3  112        J3  19 

A2        Bl  D6  H2      114        J4  29 

BIO  Personal    investments   such   as   life   insurance,   real   estate,   stocks   and    bonds. 
(Stocks  include  investments  in  building  and  loan  associations.) 
a.  Life  insurance. 

1.   To  find  the  premium  on  a  life  insurance  policy,  given  the  table  of  annual 
premiums  based  on  $1000.00,  the  kind  of  policy  and  time. 
A18  Dl        E3        F4        G6  19        Jl  42 

A21        B2        C6        Dl        E3        F5        G6  111        Jl  56 

[80] 


2.  To  find  the  difference  in  the  amount  paid  in  and  the  amount  received, 
given  the  age  at  which  the  policy  was  taken,  age  at  maturity,  kind  of 
policy,  and  table  of  premiums. 

B5  El        F4        G3  13 

B7  El        F4        G3  15 

3.  To  find  the  cost  of  protection,  given  the  face  value  of  the  policy,  the 
premium  per  year,  the  number  ot  years,  and  cash  surrender  value. 

G2  2 

G2  2 

b.  Real  estate. 

1.  To  find  the  profit,  given  the  original  cost,  the  selling  price,  other  necessary 
costs,  receipts,  and  the  time. 

D2  F2  H2  6 

D6        El        F2  H2  11 

2.  To  find  the  loss,  given  the  original  cost,  the  selling  price,  other  necessary 
costs,  losses,  receipts,  and  the  time. 

Al  El  2 

3.  To  find  the  rate  of  profit  or  loss,  given  the  cost  price,  the  selling  price,  and 
expenses.  (Selling  price  includes  receipts.  Expense  includes  added  cost 
or  losses.) 

Bl  Dl        E2        Fl  5 

B2  D6        E3        F2  13 

4.  To  find  the  cost  per  front  foot,  given  the  total  cost  and  teet  of  frontage  on 
street. 

El  1 

E4  4 

5.  To  find  the  rate  of  profit,  given  the  cost,  the  rent,  and  the  expenses  and 
losses. 

B3         D5    E2    Fl    Gl         II         13 
A3    B3    CI    D6    E2    F3    Gl         12         21 

6.  To  find  the  amount  of  rent  necessary  to  make  a  given  rate  on  an  invest- 
ment, given  the  amount  of  the  investment,  and  the  expenses. 

Al        B2        CI        Dl        El  G2  11 

Al        B2        CI        D4        E3        Fl        G2  14 

7.  To  find  the  total  price,  given  the  rent,  expenses,  and  rate  of  profit  on  the 
investment. 

El  1 

E2  2 

8.  To  find  the  net  income  on  an  investment,  given  the  amount  invested,  the 
profit,  and  the  expenses. 

Dl  Fl  2 

c.  Stocks  and  bonds. 

1.  To  find  the  dividend,  given  the  amount  of  the  bonds,  or  stock,  the  interest 
period,  and  the  rate  ot  interest. 

AlO        Bl         C3      D15      E14  G18      HIO        17        J8  86 

A16        B2        C4      D20      E16        Fl       G28      H55        19      J18        169 

2.  To  find  the  dividend,  given  the  total  cost  of  bonds,  rate  of  dividend,  and 
the   quotation. 

Gl  J2  3 

Gl  J2  3 

3.  To  find  the  amount  of  dividend,  given  the  number  of  shares  or  bonds,  the 
par  value  per  share  or  bond,  the  rate  of  dividend,  and  time. 

A3  D14        E5  12  24 

A4  CI       D14        E5  H7        13  34 

4.  To  find  the  cost  of  bonds  or  stock,  given  the  quotation,  par  value  per  share, 
brokerage,  and  number  of  shares. 

C3      D23      E14  G6      H15        II      J14         76 

Al  C3      D23      E15  G6      H15        II      J17  81 

[81] 


5.  To  find  the  cost  of  a  bond  or  bonds,  given  the  amount  of  the  bond  or  bonds, 
the  interest,  the  time,  quotation,  and  brokerage. 

a:  G1  J3  6 

a:  Gil  J3  16 

6.  To  find  the  cost  of  bonds,  giv-en  the  dividend,  the  rate  of  dividend,  broker- 
age, and  rate  of  premium. 

Gl  1 

Gl  1 

7.  To  find  the  total  cost  or  amount  of  stock,  given  the  number  of  shares  and 
quotation  or  par  value  per  share. 

A6        Bl         C:        D:  H3        19      JIO  33 

A8        Bl         C3        D9        E2  H12      110      J30  75 

8.  (a)  To  find  the  profit  or  loss,  given  the  number  of  shares  of  stock,  the 

brokerage,  the  quotation  at  which  it  was  bought,  and  the  quotation 

at  which  it  was  sold. 
A4  C3        D6        E2  G2  16         Tl  24 

A4  C3        D6        E2  G2        H2        16        jl  26 

(b)  To  find  the  profit  or  loss,  given  the  number  of  shares  of  stock,  the 

brokerage,  the  quotation  at  which  it  was  bought,  the  quotation  at 

which  it  was  sold,  and  the  rate  ot  dividend  received. 

Al  CI         Dl         EI  Gl  5 

Al  CI         Dl         E2  Gl  6 

9.  To  find  the  profit,  given  a  cost  of  stock  or  bonds,  the  amount  of  dividend 
received,  and  the  selling  price. 

HI  1 

HI  1 

10.  To  find  the  number  of  shares  or  bonds,  given  the  amount  of  dividend,  the 
rate  of  interest,  and  the  par  value  per  share  or  bond. 

Al  G2  T4 

Al  G2  "19  12 

11.  To  find  the  number  of  shares  or  bonds,  given  the  total  cost  of  stocks  or 
bonds,  the  quotation,  and  brokerage. 

C3  T7  10 

Al  C3  G2  J8  14 

12.  To  find  the  number  of  shares  or  bonds,  given  the  total  cost  or  total  amount 
of  bonds,  and  the  quotation,  or  par  value  per  share  or  bond. 

CI         D3  13         T5  12 

A2  CI        D3        El  H4        13      J"l9         33 

13.  To  find  the  amount  received  for  bonds  or  stocks,  given  the  amount  of 
bonds  or  stocks,  the  quotation  and  brokerage. 

G6  ]5  11 

G6  Jll  17 

14.  To  find  the  amount  of  bonds  or  stocks,  given  the  percent  of  dividend,  and 

the  amount  of  the  dividend. 

Bl  El  G3  II         14  10 

Bl  EI  G3  II        j9  15 

15.  To  find  the  amount  of  stocks  or  bonds,  given  the  total  cost,  the  quotation, 
and  brokerage. 

T2  2 

J3  3 

16.  To  find  the  amount  received  (amount  remitted  by  agent  after  deducting 
his  brokerage)  given  the  number  of  shares  or  bonds,  the  quotation,  and 
brokerage. 

CI         D4        E2  II  8 

A2  CI         D4        E2  II  10 

[82] 


17.  To  find  the  percent  of  profit,  given  the  quotation,  percent  of  dividend, 
brokerage  and  time. 

Bl        C6  F20        G4  14  35 

Bl        C6  F20        G4  14  35 

18.  To  find  the  rate  of  profit,  given  the  total  cost  of  bonds  or  stocks,  and  the 
amount  of  profit. 

Al  H2  Jl  4 

A2  El        Fl  H8  J3  15 

19.  To  find  the  percent  of  profit  or  loss,  given  the  amount  of  profit  or  loss,  and 
the  amount  invested. 

CI  El  2 

20.  To  find  the  rate  of  dividend,  given  the  amount  ot  dividend,  and  the  amount 
ot  bonds  or  stocks. 

Bl  Gl        HI  Jl  4 

A2        Bl  Gl        H2        12        J3  11 

21.  To  find  the  amount  of  brokerage,  given  the  total  cost  of  bonds  or  stock,  and 
the  rate  of  brokerage. 

HI  1 

HI  J2  3 

22.  To  find  the  cost  or  par  value  per  share  or  bond,  given  the  total  cost  and 
the  number  of  shares  or  bonds. 

C3  H2  Jl  6 

C3  H2  J2  7 

23.  To  find  the  proceeds,  given  the  quotation,  rate  of  interest,  and  brokerage 

A9        B2  D4  15 

A9        B2  D4  15 

Bll   Personal  activities  involving  wages  and  salaries. 
a.   Wages. 

1.  To  find  the  amount  of  wages: 

(a)  given  the  number  of  units,  and  the  wage  per  unit. 

A6      Bll        CI        D6      E16        F3        G3      H32        15      J14         97 
A50      B37      C62      D27      E47      F22      G24      H95      153      J38       455 

(b)  given  the  price  per  unit  for  a  given  number  of  units,  a  higher  price  for 
added  units,  and  a  still  higher  price  for  more  added  units,  and  the  total 
number  of  units. 

HI  1 

Bl  CI  E3  H7  12 

(c)  To  find  the  amount  of  wages  earned  in  a  given  time,  given  an  amount 
earned  in  a  different  length  of  time  at  the  same  rate. 

ElO  10 

ElO  10 

2.  To  find  the  wages  earned  per  unit,  given  the  number  of  units  and  the  total 
wage. 

Al        Bl  D3      Ell        Fl        G5        H4        II        J6         33 

A8        B7  D5      E18        Fl      Gil        H8        14        J7         69 

3.  To  find  the  number  of  units,  given  the  total  wages  earned,  and  the  wage 
per  unit. 

Al        B3  G3        H2  12  11 

Al        B3  G3        H2  J4  13 

4.  To  find  how  much  a  group  of  men  can  earn  in  a  given  time  at  a  given  rate 
per  hour,  given  a  different  number  of  men,  and  the  total  amount  earned 
at  the  same  rate  per  hour. 

D2  2 

D2  2 

5.  To  find  the  wages  earned  by  each  person,  given  the  total  wages  earneil  by 
the  total  number  ot  persons,  and  the  time  each  worked. 

D4  12  6 

U4  12  6 

[83] 


6.  To  find  the  amount  of  advance  or  decrease,  given  an  original  wage  or  pay- 
roll, and  the  percent  of  advance  or  decrease. 

D4  4 

Al        Bl  D4        El  7 

7.  To  find  the  wage  or  amount  of  payroll,  given  an  original  wage  or  an 
original  payroll,  and  the  percent  of  advance. 

B4        D7  G2    H6  19 

B5        D7    El        G2    H6        Jl    22 

8.  To  find  the  smallest  number  of  coins  and  bills  necessary  for  a  payroll, 
given  a  number  of  workmen,  the  wage  per  unit,  and  the  number  of  units. 

E2  2 

E2  2 

9.  To  find  the  rate  of  advance  or  rate  of  reduction,  given  an  original  wage, 
and  the  wage  after  the  advance  or  reduction. 

A4  D8  12 

A4  D8  12 

b.  Salaries. 

1.  To  find  a  salary,  given  an  original  salary  and  rate  of  increase. 

Bl  Dl  2 

Al        Bl  D4  15  11 

2.  To  find  the  total  salary,  given  the  number  of  units,  and  the  salary  per  unit. 

HI  1 

CI        D2        E2        Fl  Hll        II        J9         27 

3.  To  find  the  salary  per  unit,  given  the  total  salary  and  the  number  of  units. 

HI        II  2 

E2  HI        II  4 

B12   Taxation,  municipal,  state,  or  national. 

a.  To  find  the  amount  of  tax: 

1.  given  the  rate  of  taxation  and  assessed  valuation,  or  quantity.  (Note: 
Duties  and  poll  tax  included  in  this  item.) 

A14      B28        CI      D50        E8      F17      G31      H16      156      J29       250 
A39      B65      C43      D77      E22      F85     037      H29      166      J55         518 

2.  given  the  real  value,  the  ratio  of  assessment  to  real  value,  and  the  rate  of 
taxation. 

B2  D3  Fl  6 

A3        B3        C2        D9  Fl  1« 

b.  To  find  the  rate,  given  the  tax  and  assessment. 

A16      Bll         CI       D12  G9  II         JS  58 

A20      B15        C2      D18        E2        F3      G12  11      J12         85 

c.  To  find  the  total  amount  of  assessment  or  quantity  taxed,  given  the  rate,  and 
amount  of  tax  or  duty. 

A2        B3  Dl        E2  Gl  12        J9         20 

M        B4  Dl        E2  Gl  12      J12         24 

B13   Determining  economy  of  two  or  more  procedures.    (This  classification  includes 
problems  involving  difference,  saving,  choice,  and  comparison.) 
a.   Difference. 

1.  To  find  the  difference  in  unit  costs,  given  different  unit  costs  of  two  com- 
modities.   (This  includes  two  qualities  of  the  same  commodity.) 

E5  5 

Al        B3  Ell  Gl        H2  18 

2.  To  find  the  difference  in  price  per  unit,  given  the  number  of  units  and  the 
total  price  of  one  quality  of  a  commodity  and  the  number  ot  units  and 
the  total  price  of  another  quality  of  the  same  commodity. 

Dl        E2  Jl  4 

Dl        E3  Jl  5 

[84] 


3.  To  find  differences  in  amount,  given  two  different  unit  costs  for  the  same 
commodity  or  different  commodities,  and  the  total  number  of  units. 

A2        B2  E8        Fl        G3        H8        12        J3         29 

A3        B3  Dl      E12        Fl        G4        H8        13        J5         40 

4.  To  find  the  difference  in  amount,  given  a  total  cash  payment,  and  a  given 
number  of  installments  at  a  given  pavment  each. 

El  '  1 

5.  To  find  the  difference  in  the  amounts  of  a  bill,  given  different  successive 
discounts,  or  different  terms. 

J2  2 

J2  2 

6.  To  find  the  difference  in  number  of  units  purchased  for  the  same  amount 
of  money,  given  the  amount  of  money  and  different  prices  for  each  of  two 
qualities  of  the  same  commodity. 

El  1 

El  1 

7.  To  find  the  difference  in  units  of  time  between  two  places,  given  two 
distances  of  different  lengths,  and  the  rate  of  travel. 

E2  2 

E2  2 

8.  To  find  the  difference  in  the  rate  of  discount,  given  the  marked  price  and 
the  selling  price  of  one  commodity  and  the  marked  price  and  the  selling 
price  of  another  quality  ot  the  same  commodity. 

El  1 

El  1 

9.  To  find  the  difference  In  the  rate  of  travel,  given  the  distance  between  two 
places,  and  the  total  time  for  each  of  two  means  of  travel. 

El  1 

El  II  2 

10.  To  find  the  difference,  given  a  selling  price  with  a  discount  and  a  different 
selling  price  with  a  different  discount. 

Al  E2  3 

Al  E2  HI  4 

11.  To  find  the  difference  in  amount  of  profit,  given  the  amount  received  for  a 
given  number  of  units  before  spraying,  and  the  amount  received  for  a 
given  number  of  units  after  spraying,  and  the  cost  of  spraying. 

D4  '  Gl  '  5 

12.  To  find  the  difference  In  amount  of  profit,  given  an  amount  of  money 
invested  in  real  estate  with  the  cost,  time,  necessary  expenses,  rent  per 
month,  and  selling  price;  and  the  same  amount  of  money,  drawing  interest 
at  a  given  rate  for  the  same  length  of  time. 

CI        D2  Gl        H2  6 

Al  CI        D3  Gl        H2  8 

13.  To  find  the  difference  in  amount  of  profit,  given  an  amount  of  stock  with 
rate  of  dividend,  and  the  same  amount  invested  in  a  bond  and  mortgage 
with  rate  of  interest. 

Dl  1 

14.  To  find  the  difference  in  interest  due,  given  an  amount  of  money,  for  a 
given  time,  at  a  rate  of  simple  interest,  and  the  same  amount  of  money, 
for  the  same  time,  at  the  same  rate  but  compounded. 

Dl        El  2 

CI        Dl        El  3 

15.  To  find  the  difference  in  amount  of  interest  due,  given  an  amount  of  money 
drawing  interest  for  a  given  time  at  a  given  rate,  compounded  at  a  given 
period;  and  the  same  amount  of  money  drawing  interest  for  the  same  time 
at  the  same  rate,  but  compounded  at  a  given  shorter  period. 

El  HI  2 

El  HI  2 

[85] 


16.  To  find  the  difference  in  amount  of  interest,  given  the  amount,  time,  rate 
of  interest;  and  the  same  amount  of  money  for  the  same  time,  but  at  a 
different  rate  of  interest. 

CI  El  H3        II  6 

CI  El  H4        II  7 

17.  To  find  the  difference  in  the  cost  of  shipping  crated  and  uncrated  articles, 
given  the  rate  of  the  crated  article  and  the  rate  of  the  same  article  shipped 
uncrated. 

El  1 

18.  To  find  the  difference  in  cost,  given  the  total  number  of  units  traveled,  the 
total  cost  by  one  method,  and  the  cost  per  unit  per  person  by  another 
method,  and  the  number  of  persons. 

Dl  1 

Dl  "     1 

19.  To  find  the  difference  in  cost  of  sending  an  amount  of  money,  given  the 
amount  sent,  the  rate  or  charges  by  one  method,  and  the  rate  or  charges  by 
another  method. 

Dl        E2  3 

A2  Dl        E2  5 

20.  To  find  the  difference  in  premiums,  given  two  buildings  of  equal  value, 
different  material,  and  different  insurance  rates. 

El  1 

21.  To  find  the  difference,  given  an  amount  of  a  ten-year  endowment  policy, 
a  twenty-year  endowment  policy  and  a  table  ot  annual  premiums. 

'   '  El  1 

El  1 

22.  To  find  the  amount  of  difference,  given  a  salary,  plus  a  commission  on 
sales  over  a  certain  amount;  or  a  higher  commission  on  all  sales  and  no 
salary,  and  the  total  amount  of  sales. 

Dl  1 

Dl  1 

23.  To  find  the  difference  in  tax,  given  rate  and  assessed  value  in  one  area  or 
at  a  given  time  and  rate  and  assessed  value  in  another  area  or  at  another 
time. 

El  HI  2 

El  HI  2 

24.  To  find  the  difference  in  wages,  given  one  wage  and  hour  per  day  schedule, 
and  another  wage  and  hour  per  day  schedule,  and  the  time. 

E3 
b.  Saving. 

1.  To  find  the  amount  saved: 

(a)  given  the  saving  per  unit  and  the  number  of  units. 

CI  E2  Gl  12 

Al        B2        CI  E6  G2        H2        17 

(b)  given  the  number  of  units  saved,  and  the  price  per  unit. 

Gl 
A3  Dl        El  G4 

(c)  given  two  different  unit  costs  and  the  number  of  units. 

Bl        D2        F2   G2   HI 
B4   C61    D3    E4   F3   G5   HI 

(d)  given  the  itemized  list  of  original  accounts  and  the  itemized  list  of  the 
same  accounts  reduced. 

Jl  1 

J2  2 

(e)  given  the  cost  per  unit,   a  smaller  cost  per  unit  for  a  larger  lot,  and  the 
number  of  units  bought. 

HI  1 

Bl  HI  Jl  3 

[86] 


Jl 

J4 

7 
25 

Jl 

1 
10 

J2 
J4 

10 

85 

(0    given  a  price  per  unit,  a  smaller  proportionate  cost  for  a  larger  lot, 

and  the  number  of  units. 

A6        B3        CI        D5        El  G4        H8        17        J3         38 

AS        B3        C6        D6        El        Fl        G6      H15        18        J3         57 

(g)   given  a  cost  price  or  selling  price,  or  total  quantity,  and  the  percent  or 

fractional  part  saved. 

G3        HI        12        J2  8 

A2        B4  E2  G6        H5        17        J2         28 

(h)  given  an  amount  of  insurance  on  a  building,  a  rate;  a  lower  rate  due  to 
installation  of  a  safety  device,  and  the  term. 

Fl  1 

El        Fl  2 

(i)    given  the  amount  of  insurance,  the  term,  a  number  of  policies  at  a 

given  rate  for  a  long  term,  and  a  larger  number  of  policies  at  a  given 

rate  for  a  shorter  term. 

E2  2 

E2  2 

(j)  given  the  cost  price  or  selling  price  of  one  or  more  articles  in  each  of 
two  invoices,  and  the  terms. 

El  G4  5 

E2  G4  6 

(k)  given  an  amount  paid  down,  an  amount  paid  in  installments,  the  time 

and   rate   of   interest;    and    the   same    amount   down,    with   different 

amounts  of  installments  for  a  different  time,  and  no  interest. 

HI  1 

HI  1 

2.  To  find  the  percent  saved,  given  the  amount  saved  and  the  basic  price. 

Al        B3      C22  Fl        G5  Jl  32 

3.  To  find  the  amount  saved  if  the  commodity  is  home  made: 

(a)  given  an  itemized  list  of  commodities  and  the  cost  per  item  without 
cost  of  labor;  and  given  the  total  cost  for  the  complete  job  including 
the  cost  of  labor. 

A6  El        G2  9 

A8    Bl    CI         E2        G2    HI        Jl    16 

(b)  given  an  itemized  list  of  commodities  and  the  cost  per  item  with  the 
cost  of  labor  allowed  for;  and  given  the  total  cost  for  the  complete  job. 

Al  El  2 

Al        Bl  El  Gl  4 

4.  To  find  the  saving  in  premium,  given  the  value  of  a  building,  the  rate  for 
a  policy  for  one  year,  a  cheaper  rate  for  more  than  one  year,  and  the  time. 

Gl  Jl  2 

Al  Fl        Gl  Jl  4 

5.  To  find  the  time  saved,  given  the  total  time  by  one  method  of  travel,  and 
the  total  time  bv  another  method  of  travel. 

Bl  1 

Bl  1 

Choice. 

1.  To  find  the  most  economical  purchase,  given  the  cost  of  a  large  unit,  a 
larger  proportionate  cost  of  a  smaller  unit,  and  a  still  larger  proportionate 
cost  of  a  still  smaller  unit,  and  the  number  of  units. 

B2  Dl  Gl        HI 

B5  Dl  Gl        HI 

2.  To  find  the  more  economical  procedure: 
(a)   given  the  selling  price  of  a  commodity,  and  the 

discount  or  two  or  more  successive  discounts. 
Al        B2  D2  Gl 

Al        B2  D2        El  G2 

[87] 


12 
12 

Jl 

7 
11 

oice 

of  a 

single 

i.' 

7 
9 

(b)  given  the  total  number  of  units,  the  cost  per  unit  of  the  whole  com- 
modity; or  the  cost  per  unit  of  one  quality  and  the  cost  per  unit  of  the 
other  remaining  quality  of  the  commodity. 

Bl  1 

Bl  1 

(c)  given  an  amount  of  money  borrowed  for  a  given  period  of  time  at  a 
given  rate  of  interest;  or  a  part  of  the  amount  borrowed  paid  in  cash 
by  the  lender,  and  the  remainder  for  the  given  time  at  a  given  rate  of 
interest. 

HI  1 

(d)  given  one  cost  and  selling  price  and  a  different  cost  and  selling  price. 

Bl  1 

Bl  1 

3.  To  find  the  better  selling  price  of  the  same  commodity,  given  the  selling 
price  and  discount  in  one  case  and  a  lower  selling  price  in  the  second  case. 

B2  Gl  3 

B2  Gl  3 

4.  To  find  the  economical  method  of  shipping,  given  the  weight  of  an  article 
and  the  rate  charged  for  shipping  in  each  of  two  or  more  modes  of  trans- 
portation. 

C4  4 

C4  4 

5.  (a)    To  find  the  better  investment,  given  the  amount  invested  in  each  case, 

the  interest  in  each  case,  and  the  time. 

02        HI  3 

Dl  G2        HI  4 

(b)  given  one  investment  with  amount  of  profit,  and  another  investment 
with  an  amount  of  profit. 

Bl  1 

(c)  given  the  amount  invested,  the  rent,  expenses,  and  time;  and  the  same 
amount  invested,  for  the  same  time  at  a  given  rate  of  interest. 

Bl        CI        Dl  3 

B2        CI        Dl  4 

6.  To  find  the  better  wage: 

(a)  given  one  wage  and  hour  per  day  schedule,  and  another  wage  under 
another  hour  per  day  schedule. 

Al  1 

Al  1 

(b)  given  a  wage  per  unit  of  time,  the  time,  and  a  lower  wage  per  unit  of 
time  for  a  longer  time,  and  the  time. 

Al  1 

Al  1 

7.  (a)   To  find  the  better  salary,  given  a  salary  per  small  unit  of  time,  and 

another  salary  per  larger  unit  of  time. 

Dl                                          HI  2 
(b)   given  one  salary  at  one  time,  another  salary  at  a  different  time,  and 

the  ratio  of  the  value  of  a  dollar  at  the  one  time  to  the  value  at  the 
other  time. 

Bl                                                                   H2  3 

B6                                                                H2  8 

8.  To  find  the  more  economical  purchase: 

(a)   given  a  number  of  units  of  one  quality  (width,  material,  etc.)  of  a 

commodity  and  the  price  per  unit;  and  a  different  number  of  units  of  a 

different  quality  of  the  same  commodity,  at  a  different  price  per  unit. 

Bl  Dl  G2  HI  5 

Bl  Dl  G2  HI  5 

[88] 


(b)   given  different  total  prices,  with  different  successive  discounts.    (May 
be  the  same  total  prices.) 

C3        D2  5 

C3        D2  5 

9.  To  find  the  better  offer,  given  an  amount  of  insurance,  the  rate  for  a  short 

term,  the  rate  for  a  longer  term,  and  the  time. 

Al  1 

Al        B3  .4 

Comparison. 

1.  To  compare  the  results  of  two  or  more  procedures,  given  the  procedures. 

A2        B4        C2        D3        El        F2      GIO      H31      111        Jl  67 

A5        B4        C2        D7        E4        F2      Gil      H33      115        Jl  84 

2.  To  compare  areas,  given  the  dimensions  of  each. 

El  G2        H3  6 

Bl  El  G2        H3  7 

3.  To  compare  costs  of  two  or  more  commodities  or  items,  given  the  cost  of 
each  per  unit  and  the  number  of  units. 

C6  Gl      HIO  17 

C6  Gl      HIO        II  18 

4.  To  compare  pairs  of  quantities  by  ratios,  given  the  pairs  of  quantities. 
(Ratios  may  be  in  whole  numbers,  percents,  or  fractions.) 

H14  Jl  15 

C3  H14  J3         20 

5.  To  find  how  many  times  as  much  it  costs  to  send  a  given  number  of  small 
money  orders,  at  a  given  rate  per  small  money  order,  than  to  send  one 
large  money  order  equivalent  to  the  small  ones,  at  a  given  rate  per  large 
money  order. 

H2  2 

H2  2 


[89] 


APPENDIX  B 

The  following  questions  concerning  functional  relationships  are 
suggested  for  recognition  as  minimum  essentials.  The  list  is  based  upon 
the  analysis  of  ten  series  of  arithmetics  but  represents  in  part  the  judg- 
ment of  the  writer.  In  each  case  the  question  is :  What  calculations 
must  be  performed  in  order  to  find  the  quantity  named,  given  the 
quantities  specified? 

Al  To  find  totals  by  addition,  given  two  or  more  items,  values,  etc. 

A2  To  find  the  difference,  given  two  items,  values,  etc. 

A3  To  find  the  amount,  or  number  needed,  given  a  magnitude  and  the  number 

of  times  It  is  to  be  taken. 

A4  To  find  the  size  of  a  part  of  a  magnitude,  given  the  magnitude  and  the 

number  of  parts  into  which  it  is  to  be  divided. 

A5  To  find  how  many  times  a  stated  quantity  is  contained  in  a  given  magni- 

tude, given  the  quantity  and  the  magnitude. 

A6a  To  find  how  many  when  reduction  ascending  is  required,  given  a  magni- 

tude expressed  In  terms  of  a  single  denomination. 

A6b  To  find  how  many  when  reduction  ascending  Is  required,  given  a  magni- 

tude expressed  In  terms  of  two  or  more  denominations. 

A7a  To  find  how  many  when  reduction  descending  Is  required,  given  a  magni- 

tude expressed  in  terms  of  a  single  denomination. 

A7b  To  find  how  many  when  reduction  descending  Is  required,  given  a  magni- 

tude expressed  In  terms  of  two  or  more  denominations. 

A8a  To  find  a  dimension,  given  the  area  of  a  rectangle  and  one  side. 

AlOa  To  find  the  area,  given  dimensions  of  a  square,  rectangle,  or  parallelo- 

gram. 

AlOb  To  find  the  area,  given  the  base  and  altitude  of  a  triangle. 

AlOc  To  find  the  area,  given  the  diameter  of  a  circle. 

Alia  To  find  the  perimeter,  given  one  side  of  any  equilateral  figure. 

Allb  To  find  the  perimeter,  given  two  adjacent  sides  of  a  rectangle  or  parallelo- 

gram. 

Allc  To  find  the  circumference,  given  the  diameter  or  radius  of  a  circle. 

A12bl  To  find  the  cubic  contents,  given  the  three  dimensions  of  a  rectangular 

solid,  such  as  room,  bin,  woodpile,  etc. 

A12b2  To  find  the  cubic  contents,  given  the  area  of  one  surface  of  a  rectangular 

solid  and  the  depth  or  altitude. 

A14  To  find  the  average,  given  a  series  of  Items. 

A15  To  find  the  ratio  of  one  number  to  another,  given  the  two  numbers. 

A16  To  find  a  part  of  a  number,  given  the  ratio  of  the  part  to  the  number,  and 

the  number. 

A18  To  divide  a  quantity  Into  parts  having  a  given  ratio,  given  the  quantity 

and  the  ratio. 

A19  To  find  a  member  of  a  ratio,  given  two  members  of  one  ratio  and  one 

member  of  another  ratio  equal  to  the  first.    (Inverse  ratio  Included.) 

A20  To  find  the  ratio  of  items  to  total,  given  a  series  of  items. 

Blal  To  find  the  total  price,  given  the  number  of  units  and  price  per  unit. 

Bla2  To  find  the  total  price,  given  the  number  of  units  and  the  price  per  unit 

of  another  denomination. 

[90] 


To  find  the  number  of  units,  given  the  total  price  and  price  per  unit. 
To  find  the  price  per  unit,  given  the  total  price  and  the  number  of  units. 
To  find  the  amount  to  be  received  for  several  items,  given  the  price  of 
each. 

To  make  change,  given  an  amount  of  money  and  the  price  of  a  commodity. 

To  find  the  margin  or  loss,  given  the  cost  price  and  the  selling  price. 

To  find  the  total  margin  or  total  loss,  given  the  unit  cost,  the  unit  selling 

price,  and  number  of  units. 
To  find  the  selling  price,  given  the  rate  of  discount  or  loss,  and  the  price. 
To  find  the  selling  price,  given   the  rate  of  advance  or  margin  and  the 

price. 

To  find  the  selling  price,  given  the  rate  of  two  or  more  successive  discounts 
and  the  price. 

To  find  the  selling  price,  given  the  price  and  the  amount  of  commission 
or  discount. 

To  find  the  amount  of  margin,  loss,  commission  or  discount,  given  the 
total  price  and  the  rate  of  margin,  loss,  commission  or  discount. 

To  find  the  rate  of  margin,  loss,  discount,  advance  or  commission,  given 
the  total  price  and  the  amount  of  margin,  loss,  discount,  advance  or 
commission. 

To  find  the  rate  of  margin,  loss,  discount,  advance,  or  commission,  given 

the  total  price  and  the  selling  price. 
To  find  the  price,  given  the  selling  price  and  the  rate  of  margin. 
To  find  the  net  profit  or  loss,  given  the  cost  price,  overhead,  and  selling 

price. 
To  find  the  net  profit  or  loss,  given  the  total  costs  and  total  receipts. 
To  find  the  net  profit  or  loss,  given  the  itemized  costs  and  total  receipts. 
To  find  what  percent  the  net  profit  is  of  the  cost  price  or  selling  price, 

given  the  net  profit,  and  the  total  receipts,  original  outlay,  or  amount 

invested. 
To  find  what  percent  the  profit  or  loss  is  of  the  cost  price  or  selling  price, 

given  the  profit  or  loss,  and  the  cost  price  or  selling  price. 
To  find  the  interest  or  discount,  given  the  amount  loaned,  the  rate  of 

interest  or  discount,  and  the  time  or  term. 

To  find  the  amount  due,  given  the  amount  loaned,  the  rate  of  interest, 
and  the  time. 

To  find  the  balance  due,  given  the  amount  loaned,  the  time  of  interest 
payments,  the  partial  payments,  the  total  time,  and  the  rate  of  inter- 
est. 

To  find  the  total  of  a  bill  or  invoice,  given  an  item  or  series  of  items,  the 

number  of  each,  the  price  of  each,  and  the  terms. 
To  find  the  balance  of  a  cash  book,  given  expenditures  and  receipts. 
To  balance  a  bank  account,  given  an  original  balance,  a  series  of  deposits, 

and  a  series  of  withdrawals. 

To  find  how  many  times  a  given  pattern,  border,  design,  or  length  is 
contained  in  a  given  length. 

To  find  the  total  number  of  units,  given  the  dimensions  of  the  unit,  and 
the  dimensions  of  the  whole. 

To  find  the  total  number  of  units,  given  the  dimensions  of  the  whole  and 
the  size  of  the  unit. 

To  find  the  total  cost  of  construction,  given  the  cost  per  unit  and  the 
number  of  units. 

To  find  the  cost  per  unit  of  construction,  given  the  total  cost  and  the 
number  of  units. 


[91] 


B7al(a)        To  find  the  distance,  given  the  time  and  the  rate. 

B7a2(a)  To  find  the  distance  traveled  per  unit  of  time,  given  the  total  distance 
and  the  total  time. 

B7a3(a)        To  find  the  time,  given  the  distance  and  the  rate. 

B7b3(a)  To  find  the  cost  of  sending  a  commodity  or  commodities  by  parcel  post, 
given  the  rate  of  the  article  for  a  given  zone,  and  the  weight. 

B7c2(a)  To  find  the  cos  t  of  a  money  order  or  draft,  given  the  amount  sent,  and  the 
rate  charged. 

B7c2(b)  To  find  the  cost  of  mailing  letters,  newspapers,  etc.,  given  the  rate  of 
postage  per  unit  and  the  number  of  units.  (Unit  may  mean  letters 
or  weight.) 

BlObl  To  find  the  profit,  given  the  original  cost,  the  selling  price,  other  necessary 

costs,  receipts,  and  the  time.    (Real  estate.) 

BlObS  To  find  the  rate  of  profit  on  real  estate,  given  the  cost,  the  rent,  and  the 

expenses  and  losses. 

B10b6  To  find  the  amount  of  rent  necessary  to  make  a  given  rate  on  an  invest- 

ment, given  the  amount  of  the  investment,  and  the  expenses. 

BlOcl  To  find  the  dividend,  given  the  amount  of  the  bonds,  or  stock,  the  inter- 

est period,  and  the  rate  of  interest. 

Bllal(a)  To  find  the  amount  of  wages,  given  the  number  of  units,  and  the  wage 
per  unit. 

Blla2  To  find  the  wages  earned  per  unit,  given  the  number  of  units  and  the 

total  wage. 

Bllb2  To  find  the  total  salary,  given  the  number  of  units,  and  the  salary  per 

unit. 

B13bl(c)  To  find  the  amount  saved,  given  two  different  unit  costs  and  the  number 
of  units. 

B13bl(0  To  find  the  amount  saved,  given  a  price  per  unit,  a  smaller  proportionate 
cost  for  a  larger  lot,  and  the  number  of  units. 

B13b3(a)  To  find  the  amount  saved  if  the  commodity  is  home  made,  given  an  item- 
ized list  ot  commodities  and  the  cost  per  item  without  cost  of  labor; 
and  given  the  total  cost  for  the  complete  job  including  the  cost  of  labor. 

B13dl  To  compare  the  results  of  two  or  more  procedures,  given  the  procedures. 


J.a:j  2  5  :c77 


[92] 


